Matrix Coefficients, Counting and Primes for Orbits of Geometrically Finite Groups

MATRIX COEFFICIENTS, COUNTING AND PRIMES FOR ORBITS OF GEOMETRICALLY FINITE GROUPS

AMIR MOHAMMADI AND HEE OH

Abstract. Let G := SO(n, 1)◦ and Γ < G be a geometrically finite Zariski dense subgroup with critical exponent δ bigger than (n − 1)/2. Under a spectral gap hypothesis on L2(Γ\G), which is always satisfied when δ > (n − 1)/2 for n = 2, 3 and when δ > n − 2 for n ≥ 4, we obtain an effective archimedean counting result for a discrete orbit of Γ in a homogeneous space H\G where H is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family {BT ⊂ H\G} of compact subsets, there exists η > 0 such that 1−η #[e]Γ ∩ BT = M(BT ) + O(M(BT ) ) for an explicit measure M on H\G which depends on Γ. We also apply the affine sieve and describe the distribution of almost primes on orbits of Γ in arithmetic settings. One of key ingredients in our approach is an effective asymptotic formula for the matrix coefficients of L2(Γ\G) that we prove by combining methods from spectral analysis, harmonic analysis and ergodic theory. We also prove exponential mixing of the frame flows with respect to the Bowen-Margulis-Sullivan measure.

Contents 1. Introduction 1 2. Matrix coefficients in L2(Γ\G) by ergodic methods 11 3. Asymptotic expansion of Matrix coefficients 12 4. Non-wandering component of Γ\ΓHat as t → ∞ 26 5. Translates of a compact piece of Γ\ΓH via thickening 32 6. Distribution of Γ\ΓHat and Transversal intersections 37 7. Effective uniform counting 45 8. Affine sieve 57 References 61

The authors are supported in part by NSF grants. 1 2 AMIR MOHAMMADI AND HEE OH

1. Introduction Let n ≥ 2 and let G be the identity component of the special orthogonal group SO(n, 1). As well known, G can be considered as the group of orienta- n tion preserving isometries of the hyperbolic space H . A discrete subgroup Γ of G is called geometrically finite if the unit neighborhood of its convex core1 has finite Riemannian volume. As any discrete subgroup admitting a finite n sided polyhedron as a fundamental domain in H is geometrically finite, this class of discrete subgroups provides a natural generalization of lattices in G. In particular, for n = 2, a discrete subgroup of G is geometrically finite if and only if it is finitely generated. In the whole introduction, let Γ be a torsion-free geometrically finite, Zariski dense, discrete subgroup of G. We denote by δ the critical exponent of Γ. Note that any discrete subgroup of G with δ > (n − 2) is Zariski dense in G. The main aim of this paper is to obtain effective counting results for discrete orbits of Γ in H\G, where H is the trivial group, a symmetric subgroup or a horospherical subgroup of G, and to discuss their applications in the affine sieve on Γ-orbits in an arithmetic setting. Our results are formulated under a suitable spectral gap hypothesis for L2(Γ\G) (see Def. 1.1 and 1.3). This hypothesis on Γ is known to be true if the critical exponent δ is strictly bigger than n − 2. Though we believe that the condition δ > (n − 1)/2 should be sufficient to guarantee this hypothesis, it is not yet known in general (see 1.2). For Γ lattices, i.e., when δ = n − 1, both the effective counting and applications to an affine sieve have been extensively studied (see [16], [17], [4], [44], [21], [42],[48], [20], etc. as well as survey articles [51], [49] [36], [37]). Hence our main focus is when Γ is of infinite co-volume in G.

1.1. Effective asymptotic of Matrix coefficients for L2(Γ\G). We begin by describing an effective asymptotic result on the matrix coefficients for L2(Γ\G), which is a key ingredient in our approach as well as of independent interest. When Γ is not a lattice, a well-known theorem of Howe and Moore 2 [27] implies that for any Ψ1, Ψ2 ∈ L (Γ\G), the matrix coefficient Z haΨ1, Ψ2i := Ψ1(ga)Ψ2(g)dg Γ\G decays to zero as a ∈ G tends to infinity (here dg is a G-invariant measure on Γ\G). Describing the precise asymptotic is much more involved. Fix a Cartan decomposition G = KAK where K is a maximal compact subgroup and A is a one-parameter subgroup of diagonalizable elements. Let M denote the centralizer of A in K. The quotient spaces G/K and G/M can n 1 n be respectively identified with H and its unit tangent bundle T (H ), and

1 n n The convex core CΓ ⊂ Γ\H of Γ is the image of the minimal convex subset of H which contains all geodesics connecting any two points in the limit set of Γ. EFFECTIVE COUNTING 3 we parameterize elements of A = {at : t ∈ R} so that the right translation 1 n action of at in G/M corresponds to the geodesic flow on T (H ) for time t. n n We let {mx : x ∈ H } and {νx : x ∈ H } be Γ-invariant conformal densities of dimensions (n − 1) and δ respectively, unique up to scalings. Each νx is a finite measure on the limit set of Γ, called the Patterson-Sullivan measure viewed from x. Let mBMS, mBR, mBR∗ and mHaar denote, respectively, the Bowen-Margulis-Sullivan measure, the Burger-Roblin measures for the expanding and the contracting horospherical foliations, and the Liouville- 1 n measure on the unit tangent bundle T (Γ\H ), all defined with respect to the fixed pair of {mx} and {νx} (see Def. 2.1). Using the identification 1 n T (Γ\H ) = Γ\G/M, we may extend these measures to right M-invariant measures on Γ\G, which we will denote by the same notation and call them the BMS, the BR, the BR∗, the Haar measures for simplicity. We note that for δ < n − 1, only the BMS measure has finite mass [52]. In order to formulate a notion of a spectral gap for L2(Γ\G), denote by Gˆ and Mˆ the unitary dual of G and M respectively. A representation (π, H) ∈ Gˆ is called tempered if for any K-finite v ∈ H, the associated matrix coefficient function g 7→ hπ(g)v, vi belongs to L2+(G) for any > 0; non-tempered otherwise. The non-tempered part of Gˆ consists of the trivial representation, and complementary series representations U(υ, s − n + 1) ˆ n−1 parameterized by υ ∈ M and s ∈ Iυ, where Iυ ⊂ ( 2 , n − 1) is an interval depending on υ. This was obtained by Hirai [26] (see also [30, Prop. 49, 50]). Moreover U(υ, s−n+1) is spherical (i.e., has a non-zero K-invariant vector) if and only if υ is the trivial representation 1; see discussion in section 3.2. By the works of Lax-Phillips [40], Patterson [54] and Sullivan [62], if n−1 2 δ > 2 , U(1, δ − n + 1) occurs as a subrepresentation of L (Γ\G) with multiplicity one, and L2(Γ\G) possesses spherical spectral gap, meaning n−1 2 2 that there exists 2 < s0 < δ such that L (Γ\G) does not weakly contain any spherical complementary series representation U(1, s−n+1), s ∈ (s0, δ). The following notion of a spectral gap concerns both the spherical and non- spherical parts of L2(Γ\G).

Deﬁnition 1.1. We say that L2(Γ\G) has a strong spectral gap if (1) L2(Γ\G) does not contain any U(υ, δ − n + 1) with υ 6= 1; n−1 2 (2) there exist 2 < s0(Γ) < δ such that L (Γ\G) does not weakly contain any U(υ, s − n + 1) with s ∈ (s0(Γ), δ) and υ ∈ Mˆ .

n−1 2 n For δ ≤ 2 , the Laplacian spectrum of L (Γ\H ) is continuous [40]; this implies that there is no spectral gap for L2(Γ\G).

2for two unitary representations π and π0 of G, π is said to be weakly contained in π0 (or π0 weakly contains π) if every diagonal matrix coeﬃcient of π can be approximated, uniformly on compact subsets, by convex combinations of diagonal matrix coeﬃcients of π0. 4 AMIR MOHAMMADI AND HEE OH

Conjecture 1.2 (Spectral gap conjecture). If Γ is a geometrically ﬁnite and n−1 2 Zariski dense subgroup of G with δ > 2 , L (Γ\G) has a strong spectral gap. If δ > (n − 1)/2 for n = 2, 3, or if δ > (n − 2) for n ≥ 4, then L2(Γ\G) has a strong spectral gap (Theorem 3.27). Our main theorems are proved under the following slightly weaker spectral gap property assumption: 2 n−1 Deﬁnition 1.3. We say that L (Γ\G) has a spectral gap if there exist 2 < s0 = s0(Γ) < δ and n0 = n0(Γ) ∈ N such that (1) the multiplicity of U(υ, δ − n + 1) contained in L2(Γ\G) is at most dim(υ)n0 for any υ ∈ Mˆ ; 2 (2) L (Γ\G) does not weakly contain any U(υ, s−n+1) with s ∈ (s0, δ) and υ ∈ Mˆ .

The pair (s0(Γ), n0(Γ)) will be referred to as the spectral gap data for Γ. In the rest of the introduction, we impose the following hypothesis on Γ: L2(Γ\G) has a spectral gap.

Theorem 1.4. There exist η0 > 0 and ` ∈ N (depending only on the spectral ∞ gap data for Γ) such that for any real-valued Ψ1, Ψ2 ∈ Cc (Γ\G), as t → ∞, Z (n−1−δ)t Haar e Ψ1(gat)Ψ2(g)dm (g) Γ\G mBR(Ψ ) · mBR∗ (Ψ ) = 1 2 + O(S (Ψ )S (Ψ )e−η0t) |mBMS| ` 1 ` 2 2 where S`(Ψi) denotes the `-th L -Sobolev norm of Ψi for each i = 1, 2.

Remark 1.5. We remark that if either Ψ1 or Ψ2 is K-invariant, then The- n−1 orem 1.4 holds for any Zariski dense Γ with δ > 2 (without the spectral gap hypothesis), as the spherical spectral gap of L2(Γ\G) is sufficient to study the matrix coefficients associated to spherical vectors. † Let Hδ denote the sum of of all complementary series representations of 2 parameter δ contained in L (Γ\G), and let Pδ denote the projection operator 2 † 2 from L (Γ\G) to Hδ. By the spectral gap hypothesis on L (Γ\G), the main work in the proof of Theorem 1.4 is to understand the asymptotic of hatPδ(Ψ1),Pδ(Ψ2)i as t → ∞. Building up on the work of Harish-Chandra on the asymptotic behavior of the Eisenstein integrals (cf. [65], [66]), we first † obtain an asymptotic formula for hatv, wi for all K-finite vectors v, w ∈ Hδ (Theorem 3.23). This extension alone does not give the formula of the leading term of hatPδ(Ψ1),Pδ(Ψ2)i in terms of functions Ψ1 and Ψ2; however, an ergodic theorem of Roblin [56] and Winter [67] enables us to identify the main term as given in Theorem 1.4. EFFECTIVE COUNTING 5

1.2. Exponential mixing of frame flows. Via the identification of the n space Γ\G with the frame bundle over the hyperbolic manifold Γ\H , the right translation action of at on Γ\G corresponds to the frame flow for time t. The BMS measure mBMS on Γ\G is known to be mixing for the frame flows ([18], [67]). We deduce the following exponential mixing from Theorem 1.4: for a compact subset Ω of Γ\G, we denote by C∞(Ω) the set of all smooth functions on Γ\G with support contained in Ω.

Theorem 1.6. There exist η0 > 0 and ` ∈ N such that for any compact ∞ subset Ω ⊂ Γ\G, and for any Ψ1, Ψ2 ∈ C (Ω), as t → ∞, Z BMS Ψ1(gat)Ψ2(g)dm (g) Γ\G mBMS(Ψ ) · mBMS(Ψ ) = 1 2 + O(S (Ψ )S (Ψ )e−η0t) |mBMS| ` 1 ` 2 where the implied constant depends only on Ω.

For Γ convex co-compact, Theorem 1.6 for Ψ1 and Ψ2 M-invariant functions holds for any δ > 0 by Stoyanov [61], based on the approach developed by Dolgopyat [14]; however when Γ has cusps, this theorem seems to be new even for n = 2.

1.3. Effective equidistribution of orthogonal translates of an H- orbit. When H is a horospherical subgroup or a symmetric subgroup of G, we can relate the asymptotic distribution of orthogonal translates of a closed orbit Γ\ΓH to the matrix coefficients of L2(Γ\G). We fix a generalized Cartan decomposition G = HAK. We parameterize A = {at} as in section 1.1, and for H horospherical, we will assume that H is the expanding horospherical subgroup for at, that is, H = {g ∈ G : atga−t → e as t → ∞}. Haar PS Let µH and µH be respectively the H-invariant measure on Γ\ΓH defined with respect to {mx} and the skinning measure on Γ\ΓH defined with respect to {νx}, introduced in [52] (cf. (4.2)). PS Theorem 1.7. Suppose that Γ\ΓH is closed and that |µH | < ∞. There exist η0 > 0 and ` ∈ N such that for any compact subset Ω ⊂ Γ\G, any Ψ ∈ C∞(Ω) and any bounded φ ∈ C∞(Γ ∩ H\H), as t → ∞, Z (n−1−δ)t Haar e Ψ(hat)φ(h)dµH (h) h∈Γ\ΓH 1 = µPS(φ)mBR(Ψ) + O(S (Ψ) · S (φ)e−η0t) |mBMS| H ` ` with the implied constant depending only on Ω. PS For H horospherical, |µH | < ∞ is automatic for Γ\ΓH closed. For H symmetric (and hence locally isomorphic to SO(k, 1) × SO(n − k)), the cri- PS terion for the finiteness of µH has been obtained in [52] (see Prop. 4.15); PS in particular, |µH | < ∞ provided δ > n − k. 6 AMIR MOHAMMADI AND HEE OH

Letting YΩ := {h ∈ (Γ ∩ H)\H : hat ∈ Ω for some t > 0}, note that Z Z Haar Haar Ψ(hat)φ(h)dµH = Ψ(hat)φ(h)dµH YΩ PS since Ψ is supported in Ω. In the case when µH is compactly supported, YΩ turns out to be a compact subset and in this case, the so-called thickening method ([17], [29]) is sufficient to deduce Theorem 1.7 from Theorem 1.4, using the wave front property introduced in [17] (see [4] for the effective PS version). The case of µH not compactly supported is much more intricate to be dealt with. Though we obtain a thick-thin decomposition of YΩ with the thick part being compact and control both the Haar measure and the skinning measure of the thin part (Theorem 4.16), the usual method of thickening the thick part does not suffice, as the error term coming from the thin part overtakes the leading term. The main reason for this phe- Haar nomenon is because we are taking the integral with respect to µH as well as multiplying the weight factor e(n−1−δ)t in the left hand side of Theorem PS 1.7, whereas the finiteness assumption is made on the skinning measure µH . PS However we are able to proceed by comparing the two measures (at)∗µH Haar and (at)∗µH via the the transversal intersections of the orbits Γ\ΓHat with the weak-stable horospherical foliations (see the proof of Theorem 6.9 for more details). In the special case of n = 2, 3 and H horospherical, Theorem 1.7 was proved in [35], [34] and [41] by a different method.

1.4. Effective counting for a discrete Γ-orbit in H\G. In this subsection, we let H be the trivial group, a horospherical subgroup or a symmetric subgroup, and assume that the orbit [e]Γ is discrete in H\G. Theorems 1.4 and 1.7 are key ingredients in understanding the asymptotic of the number #([e]Γ ∩ BT ) for a given family {BT ⊂ H\G} of growing compact subsets, as observed in [16]. Γ We will first describe a Borel measure MH\G = MH \G on H\G, depend- n ing on Γ, which turns out to describe the distribution of [e]Γ. Let o ∈ H be 1 n + − n the point fixed by K, X0 ∈ T (H ) the vector fixed by M, X0 ,X0 ∈ ∂(H ) the forward and the backward endpoints of X0 by the geodesic flow, respec- n tively and νo the Patterson-Sullivan measure on ∂(H ) supported on the limit set of Γ, viewed from o. Let dm denote the probability Haar measure of M. Definition 1.8. For H the trivial subgroup {e}, define a Borel measure Γ MG = MG on G as follows: for ψ ∈ Cc(G),

MG(ψ) := Z 1 δt + −1 − |mBMS| ψ(k1atmk2)e dνo(k1X0 )dtdmdνo(k2 X0 ). (K/M)×A+×M×(M\K) EFFECTIVE COUNTING 7

Deﬁnition 1.9. For H horospherical or symmetric, we have either G = HA+K or G = HA+K ∪ HA−K (as a disjoint union except for the identity ± element) where A = {a±t : t ≥ 0}. Γ Deﬁne a Borel measure MH\G = MH\G on H\G as follows: for ψ ∈ Cc(H\G),

Definition 1.10. For a family {BT ⊂ H\G} of compact subsets with MH\G(BT ) tending to infinity as T → ∞, we say that {BT } is effectively well-rounded with respect to Γ if there exists p > 0 such that for all small > 0 and large T 1: + − p MH\G(BT, − BT,) = O( ·MH\G(BT )) + − + where BT, = GBT G and BT, = ∩g1,g2∈G g1BT g2 if H = {e}; and BT, = − BT G and BT, = ∩g∈G BT g if H is horospherical or symmetric. Here G denotes a symmetric -neighborhood of e in G with respect to a left invariant Riemannian metric on G. Since any two left-invariant Riemannian metrics on G are Lipschitz equivalent to each other, the above definition is independent of the choice of a Riemannian metric used in the definition of G. See Propositions 7.11, 7.15 and 7.17 for examples of effectively well- rounded families. For instance, if G acts linearly from the right on a finite dimensional linear space V and H is the stabilizer of w0 ∈ V , then the family of norm balls BT := {Hg ∈ H\G : kw0gk < T } is effectively well-rounded. If Γ is a lattice in G, then MH\G is essentially the leading term of the invariant measure in H\G and hence the definition 1.10 is equivalent to the one given in [4], which is an effective version of the well-roundedness condition given in [17]. Under the additional assumption that H ∩ Γ is a lattice in H, it is known that if {BT } is effectively well-rounded, then

1−η0 #([e]Γ ∩ BT ) = Vol(BT ) + O(Vol(BT ) ) (1.11) for some η0 > 0, where Vol is computed with respect to a suitably normalized invariant measure on H\G (cf. [16], [17], [44], [20], [4]). We present a generalization of (1.11). In the next two theorems 1.12 and 1.14, we let {Γd : d ∈ I} be a family of subgroups of Γ of ﬁnite index such that Γd ∩ H = Γ ∩ H. We assume that {Γd : d ∈ I} has a uniform spectral gap in the sense that supd s0(Γd) < δ and supd n0(Γd) < ∞. 8 AMIR MOHAMMADI AND HEE OH

For our intended application to the affine sieve, we formulate our effective results uniformly for all Γd’s. Theorem 1.12. Let H be the trivial group, a horospherical subgroup or a PS symmetric subgroup. When H is symmetric, we also assume that |µH | < ∞. If {BT } is effectively well-rounded with respect to Γ, then there exists η0 > 0 such that for any d ∈ I and for any γ0 ∈ Γ,

1 1−η0 #([e]Γdγ0 ∩ BT ) = M (BT ) + O(M (BT ) ) [Γ:Γd] H\G H\G Γ where MH\G = MH\G and the implied constant is independent of Γd and γ0 ∈ Γ. See Corollaries 7.16 and 7.18 where we have applied Theorem 1.12 to sectors and norm balls. Remark 1.13. Theorem 1.12 can be used to provide an effective version of circle-counting theorems studied in [35], [41], [53] and [50] (as well as its higher dimensional analogues for sphere packings discussed in [51]). We also formulate our counting statements for bisectors in the HAK decomposition, motivated by recent applications in [8] and [9]. Let τ1 ∈ ∞ ∞ τ1,τ2 ∞ Cc (H) and τ2 ∈ C (K), and define ξT ∈ C (G) as follows: for g = hak ∈ HA+K, Z τ1,τ2 −1 ξT (g) = χA+ (a) · τ1(hm)τ2(m k)dH∩M (m) T H∩M + where χ + denotes the characteristic function of A = {at : 0 < t < log T } AT T 0 0 and dH∩M is the probability Haar measure of H ∩ M. Since hak = h ak 0 −1 0 τ1,τ2 implies that h = h m and k = m k for some m ∈ H ∩ M, ξT is well- defined.

Theorem 1.14. There exist η0 > 0 and ` ∈ N such that for any compact ∞ ∞ subset H0 of H which injects to Γ\G, any τ1 ∈ C (H0), τ2 ∈ C (K), any γ0 ∈ Γ and any d ∈ I, µ˜PS(τ ) · ν∗(τ ) X τ1,τ2 H 1 o 2 δ δ−η0 ξT (γ) = BMS T + O(S`(τ1)S`(τ2)T ) δ · [Γ : Γd] · |m | γ∈Γdγ0 ∗ R R −1 − PS where νo (τ2) = K M τ2(mk)dmdνo(k X0 ) and µ˜H is the skinning measure on H with respect to Γ and the implied constant depends only on Γ and H0.

Theorem 1.14 also holds when τ1 and τ2 are characteristic functions of the so-called admissible subsets (see Corollary 7.21 and Proposition 7.11). We remark that unless H = K, Corollary 7.16, which is a special case of Theorem 1.12 for sectors in H\G, does not follow from Theorem 1.14, as the latter deals only with compactly supported functions τ1. For H = K, Theorem 1.14 was earlier shown in [7] and [64] for n = 2, 3 respectively. EFFECTIVE COUNTING 9

Remark 1.15. Non-eﬀective versions of Theorems 1.7, 1.12, and 1.14 were obtained in [52] for a more general class of discrete groups, that is, any non-elementary discrete subgroup admitting ﬁnite BMS-measure.

1.5. Application to Affine sieve. One of the main applications of Theo- rem 1.12 can be found in connection with Diophantine problems on orbits of Γ. Let G be a Q-form of G, that is, G is a connected algebraic group ◦ defined over Q such that G = G(R) . Let G act on an affine space V via a Q-rational representation in a way that G(Z) preserves V (Z). Fix a non-zero vector w0 ∈ V (Z) and denote by H its stabilizer subgroup and set H = H(R). We consider one of the following situations: (1) H is a symmetric subgroup of G or the trivial subgroup; (2) w0G ∪ {0} is Zariski closed and H is a compact extension of a horospherical subgroup of G. In the case (1), w0G is automatically Zariski closed by [23]. Set W := w0G and w0G ∪ {0} respectively for (1) and (2). Let Γ be a geometrically finite and Zariski dense subgroup of G with n−1 δ > 2 , which is contained in G(Z). If H is symmetric, we assume that PS |µH | < ∞. For a positive integer d, we denote by Γd the congruence subgroup of Γ which consists of γ ∈ Γ such that γ ≡ e mod d. For the next two theorems 1.16 and 1.17 we assume that there exists a finite set S of primes that the family {Γd : d is square-free with no prime factors in S} has a uniform spectral gap. This property always holds if δ > (n − 1)/2 for n = 2, 3 and if δ > n − 2 for n ≥ 4 via the recent works of Bourgain,Gamburd, Sarnak ([6], [5]) and Salehi-Golsefidy and Varju [58] together with the classification of the unitary dual of G (see Theorem 8.2). Let F ∈ Q[W ] be an integer-valued polynomial on the orbit w0Γ. Salehi- Golsefidy and Sarnak [57], generalizing [6], showed that for some R > 1, the set of x ∈ w0Γ with F (x) having at most R prime factors is Zariski dense in w0G. The following are quantitative versions: Letting F = F1F2 ··· Fr be a factorization into irreducible polynomials in Q[W ], assume that all Fj’s are irreducible in C[W ] and integral on w0Γ. Let {BT ⊂ w0G : T 1} be an effectively well-rounded family of subsets with respect to Γ. Theorem 1.16 (Upper bound for primes). For all T 1,

Mw0G(BT ) {x ∈ w0Γ ∩ BT : Fj(x) is prime for j = 1, ··· , r} r . (log Mw0G(BT )) Theorem 1.17 (Lower bound for almost primes). Assume further that β maxx∈BT kxk Mw0G(BT ) for some β > 0, where k · k is any norm on V . Then there exists R = R(F, w0Γ, β) ≥ 1 such that for all T 1,

Mw0G(BT ) {x ∈ w0Γ ∩ BT : F (x) has at most R prime factors} r . (log Mw0G(BT )) 10 AMIR MOHAMMADI AND HEE OH

Observe that these theorems provide a description of the asymptotic distribution of almost prime vectors, as BT can be taken arbitrarily.

Remark 1.18. In both theorems above, if all BT are K-invariant subsets, our hypothesis on the uniform spectral gap for the family {Γd} can be dis- posed again, as the uniform spherical spectral gap property proved in [58] and [6] is suﬃcient in this case. For instance, Theorems 1.16 and 1.17 can be applied to the norm balls δ/λ BT = {x ∈ w0G : kxk < T } and in this case Mw0G(BT ) T where λ denotes the log of the largest eigenvalue of a1 on the R-span of w0G. In order to present a concrete example, we consider an integral quadratic form Q(x1, ··· , xn+1) of signature (n, 1) and for an integer s ∈ Z, denote by WQ,s the aﬃne quadric given by {x : Q(x) = s}.

As well-known, WQ,s is a one-sheeted hyperboloid if s > 0, a two-sheeted hyperboloid if s < 0 and a cone if s = 0. We will assume that Q(x) = s has a non-zero integral solution, so pick w0 ∈ WQ,s(Z). If s 6= 0, the stabilizer subgroup Gw0 is symmetric; more precisely, locally isomorphic to

SO(n − 1, 1) (if s > 0) or SO(n) (if s < 0) and if s = 0, Gw0 is a compact extension of a horospherical subgroup. By the remark following Theorem PS 1.7, the skinning measure µ is finite if n ≥ 3. For n = 2 and s > 0, Gw0 is Gw0 a one-dimensional subgroup consisting of diagonalizable elements, and µPS Gw0 2 is infinite only when the geodesic in H stabilized by Gw0 is divergent and 2 goes into a cusp of a fundamental domain of Γ in H ; in this case, we call w0 externally Γ-parabolic, following [52]. Therefore the following are special cases of Theorems 1.16 and 1.17: Corollary 1.19. Let Γ be a geometrically finite and Zariski dense subgroup n−1 of SOQ(Z) with δ > 2 . In the case when n = 2 and s > 0, we additionally assume that w0 is not externally Γ-parabolic. Fixing a K-invariant norm n+1 k · k on R , we have, for any 1 ≤ r ≤ n + 1, T δ (1) {x ∈ w0Γ: kxk < T, xj is prime for all j = 1, ··· , r} (log T )r ; (2) for some R > 1, T δ {x ∈ w Γ: kxk < T, x ··· x has at most R prime factors} . 0 1 r (log T )r The upper bound in (1) is sharp up to a multiplicative constant. The lower bound in (2) can also be stated for admissible sectors under the uniform spectral gap hypothesis (cf. Corollary 7.18). Corollary 1.19 was previously obtained in cases when n = 2, 3 and s ≤ 0 ([5], [33], [35],[34], [41]). To explain how Theorems 1.16 and 1.17 follow from Theorem 1.12, let

Γw0 (d) = {γ ∈ Γ: w0γ = w0 mod (d)} for each square-free integer d. Then Stab (w ) = Stab (w ) and the family {Γ (d)} admits a uni- Γw0 (d) 0 Γ 0 w0 form spectral gap property as Γd < Γw0 (d). Hence Theorem 1.12 holds EFFECTIVE COUNTING 11

for the congruence family {Γw0 (d): d is square-free, with no small primes}, providing a key axiomatic condition in executing the combinatorial sieve (see [28, 6.1-6.4], [25, Theorem 7.4], as well as [6, Sec. 3]). When an explicit uniform spectral gap for {Γd} is known (e.g., [19], [43]), the number R(F, w0Γ) can also be explicitly computed in principle. The paper is organized as follows. In section 2, we recall the ergodic result of Roblin which gives the leading term of the matrix coefficients for L2(Γ\G). In section 3, we obtain an effective asymptotic expansion for the matrix coefficients of the complementary series representations of G (Theorem 3.23) as well as for those of L2(Γ\G), proving Theorem 1.4. In section 4, we study the reduction theory for the non-wandering component of Γ\ΓHat, describing its thick-thin decomposition; this is needed as Γ\ΓH has infinite Haar-volume in general. We will see that the non-trivial dynamics of Γ\ΓHat as t → ∞ can be seen only within a subset of finite PS-measure. In section 5, for φ compactly supported, we prove Theorem 1.7 using Theorem 1.4 via thickening. For a general bounded φ, Theorem 1.7 is obtained via a careful study of the transversal intersections in section 6. Theorem 1.6 is also proved in section 6. Counting theorems 1.12 and 1.14 are proved in section 7 and Sieve theorems 1.16 and 1.17 are proved in the final section 8.

2. Matrix coefficients in L2(Γ\G) by ergodic methods ◦ + n Throughout the paper, let G be SO(n, 1) = Isom (H ) for n ≥ 2, i.e., n the group of orientation preserving isometries of (H , d), and Γ < G be a n non-elementary torsion-free geometrically finite group. Let ∂(H ) denote n n the geometric boundary of H . Let Λ(Γ) ⊂ ∂(H ) denote the limit set of Γ, and δ the critical exponent of Γ, which is known to be equal to the Hausdorff dimension of Λ(Γ) [63]. n A family of measures {µx : x ∈ H } is called a Γ-invariant conformal n density of dimension δµ > 0 on ∂(H ), if each µx is a non-zero finite Borel n n n measure on ∂(H ) satisfying for any x, y ∈ H , ξ ∈ ∂(H ) and γ ∈ Γ, dµ y −δµβξ(y,x) γ∗µx = µγx and (ξ) = e , dµx −1 n where γ∗µx(F ) = µx(γ (F )) for any Borel subset F of ∂(H ). Here βξ(y, x) denotes the Busemann function: βξ(y, x) = limt→∞ d(ξt, y) − d(ξt, x) where ξt is a geodesic ray tending to ξ as t → ∞. We denote by {νx} the Patterson-Sullivan density, i.e., a Γ-invariant con- n formal density of dimension δ and by {mx : x ∈ H } a Lebesgue density, n i.e., a G-invariant conformal density on the boundary ∂(H ) of dimension (n − 1). Both densities are determined unique up to scalar multiples. t 1 n 1 n Denote by {G : t ∈ R} the geodesic flow on T (H ). For u ∈ T (H ), ± n we denote by u ∈ ∂(H ) the forward and the backward endpoints of the ± t n geodesic determined by u, i.e., u = limt→±∞ G (u). Fixing o ∈ H , the 12 AMIR MOHAMMADI AND HEE OH map + − u 7→ (u , u , s = βu− (o, π(u))) 1 n is a homeomorphism between T (H ) with n n n (∂(H ) × ∂(H ) − {(ξ, ξ): ξ ∈ ∂(H )}) × R. Using this homeomorphism, we define measuresm ˜ BMS, m˜ BR, m˜ BR∗ ,m ˜ Haar 1 n on T (H ) as follows ([11], [45], [63], [12], [56]): Definition 2.1. Set BMS δβ (o,π(u)) δβ (o,π(u)) + − (1) dm˜ (u) = e u+ e u− dνo(u )dνo(u )ds; BR (n−1)β (o,π(u)) δβ (o,π(u)) + − (2) dm˜ (u) = e u+ e u− dmo(u )dνo(u )ds; BR∗ δβ (o,π(u)) (n−1)β (o,π(u)) + − (3) dm˜ (u) = e u+ e u− dνo(u )dmo(u )ds; Haar (n−1)β (o,π(u)) (n−1)β (o,π(u)) + − (4) dm˜ (u) = e u+ e u− dmo(u )dmo(u )ds.

The conformal properties of {νx} and {mx} imply that these definitions n are independent of the choice of o ∈ H . We will extend these measures to n 1 n G; these extensions depend on the choice of o ∈ H and X0 ∈ To(H ). Let n 1 n K := StabG(o) and M := StabG(X0), so that H ' G/K and T (H ) ' G/M. Let A = {at : t ∈ R} be the one-parameter subgroup of diagonalizable t elements in the centralizer of M in G such that G (X0) = [M]at = [atM]. 1 n Using the identification T (H ) with G/M, we lift the above measures to G, which will be denoted by the same notation by abuse of notation, so that they are all invariant under M from the right. These measures are all left Γ-invariant, and hence induce locally finite Borel measures on Γ\G, which we denote by mBMS (the BMS-measure), mBR BR∗ Haar (the BR-measure), m (the BR∗ measure), m (the Haar measure) by abuse of notation. Let N + and N − denote the expanding and the contracting horospherical subgroups, i.e., ± N = {g ∈ G : atga−t → e as t → ±∞}. For g ∈ G, define ± ± n g := (gM) ∈ ∂(H ). We note that mBMS, mBR, and mBR∗ are invariant under A, N + and N − respectively and their supports are given respectively by {g ∈ Γ\G : g+, g− ∈ Λ(Γ)}, {g ∈ Γ\G : g− ∈ Λ(Γ)} and {g ∈ Γ\G : g+ ∈ Λ(Γ)}. The measure mHaar is invariant under both N + and N −, and hence under G, as N + and N − generate G topologically. That is, mHaar is a Haar measure of G. We consider the action of G on L2(Γ\G, mHaar) by right translations, which gives rise to the unitary action for the inner product: Z Haar hΨ1, Ψ2i = Ψ1(g)Ψ2(g)dm (g). Γ\G EFFECTIVE COUNTING 13

Theorem 2.2. Let Γ be Zariski dense. For any functions Ψ1, Ψ2 ∈ Cc(Γ\G),

BR BR∗ (n−1−δ)t m (Ψ1) · m (Ψ2) lim e hatΨ1, Ψ2i = . t→∞ |mBMS|

Proof. Roblin [56] proved this for M-invariant functions Ψ1 and Ψ2. His 1 n proof is based on the mixing of the geodesic flow on T (Γ\H ) = Γ\G/M. For Γ Zariski dense, the mixing of mBMS was extended to the frame flow on Γ\G, by [67]. Based on this, the proof given in [56] can be repeated verbatim to prove the claim (cf. [67]). 3. Asymptotic expansion of Matrix coefficients 3.1. Unitary dual of G. Let G = SO(n, 1)◦ for n ≥ 2 and K a maximal compact subgroup of G. Denoting by g and k the Lie algebras of G and K respectively, let g = k ⊕ p be the corresponding Cartan decomposition of g. Let A = exp(a) where a is a maximal abelian subspace of p and let M be the centralizer of A in K. Define the symmetric bi-linear form h·, ·i on g by 1 hX,Y i := B(X,Y ) (3.1) 2(n − 1) where B(X,Y ) = Tr(adXadY ) denotes the Killing form for g. The reason n for this normalization is so that the Riemannian metric on G/K ' H induced by h·, ·i has constant curvature −1. ij Let {Xi} be a basis for gC over C; put gij = hXi,Xji and let g be the (i, j) entry of the inverse matrix of (gij). The element X ij C = g XiXj is called the Casimir element of gC (with respect to h·, ·i). It is well-known that this definition is independent of the choice of a basis and that C lies in the center of the universal enveloping algebra U(gC) of gC. Denote by Gˆ the unitary dual, i.e., the set of equivalence classes of irreducible unitary representations of G. A representation π ∈ Gˆ is said to be tempered if for any K-finite vectors v1, v2 of π, the matrix coefficient func- 2+ tion g 7→ hπ(g)v1, v2i belongs to L (G) for any > 0. We describe the non-tempered part of Gˆ in the next subsection. 3.2. Standard representations and complementary series. Let α denote the simple relative root for (g, a). The root subspace n of α has dimension n − 1 and hence ρ, the half-sum of all positive roots of (g, a) with n−1 multiplicities, is given by 2 α. Set N = exp n. By the Iwasawa decomposition, every element g ∈ G can be written uniquely as g = kan with k ∈ K, a ∈ A and n ∈ N. We write κ(g) = k, exp H(g) = a and n(g) = n. For any g ∈ G and k ∈ K, we let κg(k) = κ(gk), and Hg(k) = H(gk) so that gk = κg(k) exp(Hg(k))n(gk). 14 AMIR MOHAMMADI AND HEE OH

Given a complex valued linear function λ on a, we deﬁne a G-representation U λ on L2(K) by the prescription: for φ ∈ L2(K) and g ∈ G,

λ (−(λ+2ρ)◦Hg−1 ) U (g)φ = e · φ ◦ κg−1 . (3.2) This is called a standard representation of G (cf. [65, Sec. 5.5]). Observe that the restriction of U λ to K coincides with the left regular representation 2 λ −1 of K on L (K): U (k1)f(k) = f(k1 k). If R denotes the right regular representation of K on L2(K), then R(m)U λ(g) = U λ(g)R(m) for all m ∈ M. In particular each M-invariant subspace of L2(K) for the right translation action is a G-invariant subspace of U λ. Following [65], for any υ ∈ Mˆ , we let Q(υ)L2(K) denote the isotypic R(M) submodule of L2(K) of type υ. Choosing a ﬁnite dimensional vector space, say, V on which M acts irreducibly via υ, it is shown in [65] that the υ-isotypic space Q(υ)L2(K) can be written as a sum of dim(υ) copies of Uυ(λ) where for each m ∈ M, U (λ) = f ∈ L2(K,V ): . υ f(km) = υ(m)f(k), for almost all k ∈ K

n−1 If λ ∈ ( + iR)α, then Uυ(λ) is unitary with respect to the inner product 2 R hf1, f2i = K hf1(k), f2(k)iV dk, and called a unitary principal series representation. These representations are tempered. A representation Uυ(λ) with n−1 λ∈ / ( 2 + iR)α is called a complementary series representation if it is uni- tarizable. For λ = rα, we will often omit α for simplicity. For n = 2, the ◦ complementary series representations of G = SO(2, 1) are U1(s − 1) with 1/2 < s < 1; in particular they are all spherical. For n ≥ 3, a representation υ ∈ Mˆ is specified by its highest weight, which can be regarded as a n−1 sequence of 2 integers with j1 ≥ j2 ≥ · · · ≥ |j(n−1)/2| if n is odd, and as n−2 a sequence of 2 integers with j1 ≥ j2 ≥ · · · ≥ j(n−2)/2 ≥ 0 if n is even. In both cases, let ` = `(υ) be the largest index such that j` 6= 0 and put `(υ) = 0 if υ is the trivial representation. Then the complementary series n−1 representations are precisely Uυ(s − n + 1), s ∈ Iυ := ( 2 , (n − 1) − `), up to equivalence. In particular, the spherical complementary series representations are ex- hausted by {U1(s − n + 1) : (n − 1)/2 < s < n − 1}. The complementary representation Uυ(λ) contains the minimal K-type, say, συ with multiplicity one. The classification of Gˆ says that if π ∈ Gˆ is non-trivial and non-tempered, then π is (equivalent to) the unique irreducible subquotient of the complementary series representation Uυ(s − n + 1), s ∈ Iυ, containing the K-type συ, which we will denote by U(υ, s − n + 1). This classification was obtained by Hirai [26]; see also [30, Prop. 49 and 50]) and [3]. Note that U(υ, s − n + 1) is spherical if and only if Uυ(s − n + 1) is spherical if and only if υ = 1. For convenience, we will call U(υ, s − n + 1) a complementary series representation of parameter (υ, s). EFFECTIVE COUNTING 15

Observe that the non-spherical complementary series representations exist n−1 only when n ≥ 4. For 2 < s < n − 1, we will set Hs := U(1, s − n + 1), i.e., the spherical complementary series representations of parameter s. Our normalization of the Casimir element C is so that C acts on Hs as the scalar s(s − n + 1). In order to study the matrix coefficients of complementary series representations, we work with the standard representations, which we first relate with Eisenstein integrals. 3.3. Generalized spherical functions and Eisenstein integrals. Fix a complex valued linear function λ on a, and the standard representation U λ. By the Peter-Weyl theorem, we may decompose the left-regular representa- 2 2 tion V = L (K) as V = ⊕σ∈Kˆ Vσ, where Vσ = L (K; σ) denotes the isotypic K-submodule of type σ, and Vσ ' dσ · σ where dσ denotes the dimension of σ. P 2 Set ΩK = 1 + ωK = 1 − Xi where {Xi} is an orthonormal basis of kC. It belongs to the center of the universal enveloping algebra of kC. By Schur’s lemma, ΩK acts on Vσ by a scalar, say, c(σ). Since ΩK acts as a skew-adjoint operator, c(σ) is real. Moreover c(σ) ≥ 1, see [65, p. 261], and ` ` kΩK vk = c(σ) kvk for any smooth vector v ∈ Vσ. Furthermore it is shown in [65, Lemma 4.4.2.3] that if ` is large enough, then X 2 −` dσ · c(σ) < ∞. (3.3) σ∈Kˆ For σ ∈ Kˆ and k ∈ K define

χσ(k) := dσ · Tr(σ(k)) where Tr is the trace operator. For any continuous representation W of K, and σ ∈ Kˆ , the projection operator from W to the σ-isotypic space W (σ) is given as follows: Z Pσ = χσ(k)W (k)dk. K

For υ ∈ Mˆ , we write υ ⊂ σ if υ occurs in σ|M , and we write υ ⊂ σ ∩ τ if υ occurs both in σ|M and τ|M . We remark that for σ ∈ Kˆ , given υ ∈ Mˆ occurs at most once in σ|M ([15]; [65, p.41]). For υ ⊂ σ, we denote by Pυ the projection operator from Vσ to the υ-isotypic subspace Vσ(υ) ' dσ · υ P so that any w ∈ Vσ can be written as w = υ⊂σ Pυ(w). By the theory of representations of compact Lie groups, we have for any f ∈ L2(K; σ) we have hf, χ¯σi = f(e). In the rest of this subsection, we ﬁx σ, τ ∈ Kˆ . Deﬁne an M-module homomorphism T0 : Vσ → Vτ by X T0(w) = hPυ(w), Pυ(¯χσ)iPυ(¯χτ ). υ⊂σ∩τ 16 AMIR MOHAMMADI AND HEE OH

Set E := HomC(Vσ,Vτ ). Then E is a (τ, σ)-double representation space, a left τ and right σ-module. We put

EM := {T ∈ E : τ(m)T = T σ(m) for all m ∈ M}. λ λ λ Denote by Uσ and Uτ the representations of K obtained by restricting U |K to the subspace Vσ and Vτ respectively. Deﬁne Tλ ∈ E by Z λ λ −1 Tλ := Uτ (m)T0Uσ (m )dm M where dm denotes the probability Haar measure of M. It is easy to check that Tλ ∈ EM . R λ λ −1 λ(H(ak)) An integral of the form K Uτ (κ(ak))TλUσ (k )e dk is called an Eisenstein integral. Clearly, the matrix coeﬃcients of the representation U λ are understood λ if we understand Pτ U (a)Pσ for all τ, σ ∈ Kˆ , which can be proved to be an Eisenstein integral: Theorem 3.4. For any a ∈ A, we have Z λ λ λ −1 λ(H(ak)) Pτ U (a)Pσ = Uτ (κ(ak))TλUσ (k )e dk. K ˆ 2 λ −1 P Proof. For • ∈ K and φ ∈ L (K; •), we write U• (k )φ = υ⊂• φk,υ, that λ −1 P is, φk,υ = Pυ(U• (k )φ). In particular, φ(k) = υ⊂• φk,υ(e) and λ −1 φk,υ(e) = hφk,υ, χ¯•i = hU• (k )φ, Pυ(¯χ•)i.

Let ϕ ∈ Vσ and ψ ∈ Vτ . For any g ∈ G, we have λ λ −1 X λ −1 λ −1 hUτ (κ(gk))T0Uσ (k )ϕ, ψi = hUσ (k )ϕ), Pυ(¯χσ)ihPυ(¯χτ ),Uτ (κ(gk)) ψi υ⊂σ∩τ X = ϕk,υ(e)ψκ(gk),υ(e). (3.5) υ⊂σ∩τ On the other hand, we have

Z −1 hU λ(a)ϕ, ψi = ϕ(κ(a−1k)ψ(k)e−(λ+2ρ)H(a k)dk K Z = ϕ(k)ψ(κ(ak))eλ(H(ak))dk K Z X X λ(H(ak)) = ( ϕk,υ(e))( ψκ(ak),υ(e))e dk. (3.6) K υ⊂σ υ⊂τ

We now claim that; if υ1 6= υ2, then Z λ(H(ak)) ϕk,υ1 (e)ψκ(ak),υ2 (e)e dk = 0 K To see this, ﬁrst note that M and a commute, and hence H(amk) = H(ak), and κ(amk) = mκ(ak). We also note that

ϕk,υ1 ∈ Vσ(υ1), and ψκ(ak),υ2 ∈ Vτ (υ2). EFFECTIVE COUNTING 17

Now if υ1 6= υ2, then by Schur’s orthogonality of matrix coeﬃcients, Z −1 −1 ϕk,υ1 (m )ψκ(ak),υ2 (m )dm = M Z λ λ 0 hUσ (m)ϕk,υ1 , Pυ1 (¯χσ)ihUτ (m)ψk ,υ2 , Pυ2 (¯χτ )idm = 0; M we get Z λ(H(ak)) ϕk,υ1 (e)ψκ(ak),υ2 (e)e dk K Z Z λ(H(amk)) = ( ϕmk,υ1 (e)ψκ(amk),υ2 (e)e dm)dk M\K M Z Z λ(H(ak)) −1 −1 = e ( ϕk,υ1 (m )ψκ(ak),υ2 (m )dm)dk = 0, M\K M implying the claim. Therefore, it follows from (3.5) and (3.6) that for any ϕ ∈ Vσ and ψ ∈ Vτ , λ λ hPτ U (a)Pσϕ, ψi = hU (a)ϕ, ψi Z X λ(H(ak)) = ϕk,υ(e)ψκ(gk),υ(e)e dk K υ⊂σ∩τ Z λ λ −1 λ(H(ak)) = hUτ (κ(ak))T0Uσ (k )ϕ, ψie dk K Z λ λ −1 λ(H(ak)) = hUτ (κ(ak))TλUσ (k )ϕ, ψie dk; K we have used κ(akm) = κ(ak)m and H(akm) = H(ak) for the last equality. It follows that Z λ λ λ −1 λ(H(ak)) Pτ U (a)Pσ = Uτ (κ(ak))TλUσ (k )e dk. K For the special case of τ = σ, this theorem was proved by Harish-Chandra (see [66, Thm. 6.2.2.4]), where T0 was taken to be T0(w) = (w, χ¯σ)¯χσ and R λ λ −1 Tλ = M Uσ (m)T0Uσ (m )dm. Lemma 3.7. For any λ ∈ C, 2 2 kTλk ≤ dσ · dτ where kTλk denotes the operator norm of Tλ. 2 ˆ Proof. Since kχυk ≤ dυ for any υ ∈ M, 2 X 2 X 4 kT0k ≤ kχυk ≤ dυ υ⊂σ∩τ υ⊂σ∩τ X 2 X 2 2 2 ≤ ( dυ · ( dυ) ≤ dσ · dτ . υ⊂σ υ⊂τ Since kTλk ≤ kT0k · kσk · kτk = kT0k, the claim follows. 18 AMIR MOHAMMADI AND HEE OH

3.4. Harish-Chandra’s expansion of Eisenstein integrals. Fix σ, τ ∈ Kˆ . Let E and EM to be as in the previous subsection. + Given T ∈ EM , r ∈ C, and at ∈ A , we investigate an Eisenstein integral: Z −1 (irα−ρ)(H(atk)) τ(κ(atk))Tirα−ρσ(k )e dk. K We recall the following fundamental result of Harish-Chandra:

Theorem 3.8. (Cf. [66, Theorem 9.1.5.1]) There exists a subset Oσ,τ of C, whose complement is a locally ﬁnite set, such that for any r ∈ Oσ,τ there exist uniquely determined functions c+(r), c−(r) ∈ HomC(EM ,EM ) such that for all T ∈ EM , Z −1 (irα−ρ)H(atk) ρ(at) τ(κ(atk))T σ(k )e dk K = Φ(r : at)c+(r)T + Φ(−r : at)c−(r)T

+ where Φ is a function on Oσ,τ ×A taking values in HomC(EM ,EM ), deﬁned as in (3.12).

Let us note that, fixing T and at, the Eisenstein integral on the left hand side of the above is an entire function of r; see [66, Section 9.1.5]. Much of the difficulties lie in the fact that the above formula holds only for Oσ,τ but not for the entire complex plane, as we have no knowledge of which complementary series representations appear in L2(Γ\G). We need to un- R −1 (sα−2ρ)H(atk) derstand the Eisenstein integral K τ(κ(atk))Tsα−2ρσ(k )e dk n−1 for every s ∈ ( 2 , n − 1). We won’t be able to have as precise as a formula as Theorem 3.8 but will be able to determine a main term with an exponential error term. We begin by discussing the definition and properties of the functions Φ and c±.

3.4.1. The function Φ. As in [66, page 287], we will recursively define rational functions {Γ` : ` ∈ Z≥0} which are holomorphic except at a locally finite subset, say Sσ,τ . The subset Oσ,τ in Theorem 3.8 is indeed C−∪r∈Sσ,τ {±r}. More precisely, let l be the Lie algebra of the Cartan subalgebra (=the centralizer of A). Let Hα ∈ lC be such that B(H,Hα) = α(H) for all H ∈ lC. Let X ∈ g±α be chosen so that [X ,X ] = H and [H,X ] = ±α C α −α α α α(H)Xα. In particular, B(Xα,X−α) = 1. Write X±α = Y±α + Z±α where Y±α ∈ kC and Z±α ∈ pC. Letting ΩM denote the Casimir element of M, given S ∈ HomC(EM ,EM ), we define f(S) by

f(S)T = ST σ(ΩM ), for all T ∈ EM . n−1 We now deﬁne Γ` := Γ`(ir − 2 )’s in Q(aC) ⊗ HomC(EM ,EM ) by the following recursive relation (see [66, p. 286] for the def. of Q(aC)): Γ0 = I EFFECTIVE COUNTING 19 and X {`(2ir − n + 1) − `(` − n + 1) − f} Γ` = ((2ir−n+1)−2(`−2j))Γ`−2j j≥1 X X +8 (2j−1)τ(Yα)σ(Y−α)Γ`−(2j−1)−8 j {τ(YαY−α) + σ(YαY−α)} Γ`−2j. j≥1 j≥1

The set Oσ,τ consists of r’s such that {`(2ir − n + 1) − `(` − n + 1) − f} is invertible so that the recursive deﬁnition of the Γ`’s is meaningful.

Lemma 3.9. Fix any t0 > 0 and a compact subset ω ⊂ Oσ,τ . There exist bω (depending only on t0 and ω) and N0 > 1 (independent of σ, τ ∈ Kˆ ) such that for any r ∈ ω and ` ∈ N,

n−1 N0 N0 `t0 kΓ`(ir − 2 )k ≤ bωdσ dτ e . Proof. Our proof uses an idea of the proof of [66, Lemmas 9.1.4.3-4]. For n−1 s = ir − 2 , and T ∈ HomC(EM ,EM ), deﬁne 2 Λ`(T ) := −` + `(2s − n + 1) − f T.

For qσ and qτ which are respectively the highest weights for σ and τ, since qσ dσ with implied constant independent of σ ∈ Kˆ ,

max{kτ(Yα)σ(Y−α)k, kτ(YαY−α)k, σ(YαY−α)k} 2 2 0 3 3 ≤ c0 · (qσqτ + qσ + qσ)dσdτ ≤ c0dσdτ 0 for some c0, c0 > 0 independent of σ and τ. Hence for some c1 = c1(ω), for all r ∈ ω,

n−1 −1 3 3 X kΓ`(ir − 2 )k ≤ ` · kΛ` k · c1dσdτ kΓ`−jk. (3.10) j<`

2 −1 −t0 −1 3 3 Let N1 be an integer such that ` · kΛ` k · (1 − e ) c1dσdτ ≤ N1 for −1 −2 all ` ≥ 1. Since kΛ` k ` as ` → ∞ and the coeﬃcients of f depend only on the eigenvalues of ΩM for those υ ∈ Mˆ contained in σ, we can take N2 N2 N1 = N1(ω) so that N1 ≤ c2dσ dτ for some N2 ≥ 1 and c2 = c2(ω) > 1 (independent of σ and τ). Set

n−1 −`t0 M(t0, ω) := max kΓ`(ir − 2 )ke . `≤N1,r∈ω

By (3.10) together with the observation that both N1 = N1(ω) and −1 max`≤N1 kΛ` k are bounded by a polynomial in dσ and dτ , we have M(t0, ω) ≤ N0 N0 bωdσ dτ for some N0 ≥ 1 and bω > 0. We shall now show by induction that for all r ∈ ω and for all ` ≥ 1,

n−1 `t0 kΓ`(ir − 2 )k ≤ M(t0, ω)e . (3.11) 20 AMIR MOHAMMADI AND HEE OH

First note that (3.11) holds for all ` ≤ N1 by the deﬁnition of M(t0, ω). Now for any N1 > N, suppose (3.11) holds for each ` < N. Then n−1 −1 2 −1 3 3 X n−1 kΓN (ir − 2 )k ≤ N (N kΛN kc1dσdτ ) kΓN−j(ir − 2 )k j

−1 −t0 X (N−j)t0 ≤ N N1(1 − e )M(t0, ω) e j

Nt0 ≤ M(t0, ω)e , ﬁnishing the proof. Following Warner (Cf. [66, Theorem 9.1.4.1]), we deﬁne irt X n−1 −`t Φ(r : at) = e Γ`(ir − 2 )e (3.12) `≥0 which converges for all large enough t by Lemma 3.9.

− − 3.4.2. The function c±. Let N = exp(n ) be the root subspace corre- − sponding to −α, and dN − denote a Haar measure on N normalized so that R −2ρ(H(n)) N − e dN − (n) = 1. The following is due to Harish-Chandra; see [66, Thm. 9.1.6.1].

Theorem 3.13. For r ∈ Oσ,τ with =(r) < 0, c+(r) is holomorphic and given by Z −1 −(irα+ρ)(H(n)) c+(r)T = T σ(κ(n) )e dN − (n). N − R −(irα+ρ)H(n) The integral N − e dN − (n) is absolutely convergent iﬀ =(r) < 0, shown by Gindikin and Karperlevic ([66, Coro.9.1.6.5]).

Corollary 3.14. For any r ∈ Oσ,τ with =(r) < 0, the operator norm R (=(r)α−ρ)H(n) kc+(r)k is bounded above by N − e dN − (n). Proof. Since the operator norm kσ(k)k is 1 for any k ∈ K, the claim is immediate from Theorem 3.13.

Proposition 3.15. Fix a compact subset ω contained in Oσ,τ ∩ {=(r) < 0}. There exist d1 = d1(ω) and N2 > 1 such that for any r ∈ ω, we have

N2 N2 kc±(r)T±irα−ρk ≤ d1 · dτ dσ . R −(<(ir)α+ρ)H(n) Proof. By the assumption on ω, the integral N − e dN − (n) converges uniformly for all r ∈ ω. Hence the bound for c+(r)Tirα−ρ follows from Corollary 3.14 together with Lemma 3.7. To get a bound for c−(r)T−irα−ρ, we recall that

n−1 Z (−ir+ )t −1 (irα−ρ)(H(atk)) e 2 τ(κ(atk))T σ(k )e dk K −irt −irt = e Φ(r : at)c+(r)T + e Φ(−r : at)c−(r)T. EFFECTIVE COUNTING 21

−irt P n−1 −`t Then e Φ(−r : at) = I+ `≥1 Γ`(−ir− 2 )e , and applying Lemma 3.9 with t0 = 1, we get

X n−1 −`t N0 N0 X `(1−t) kΓ`(−ir − 2 )e k ≤ bωdσ dτ e . `≥1 `≥1

N0 N0 P `(1−t0) Fix t0 > 0 so that bωdσ dτ `≥1 e < 1/2; then t0 log(dσdτ ). Now −irt0 0 Ar := e Φ(−r : at0 ) is invertible and for some N1 and bω,

−1 0 N1 N1 kAr k ≤ bωdσ dτ . (3.16)

R (irα−ρ)(H(at0 k)) Since the map k 7→ H(at0 k) is continuous, we have K |e |dk < dω for all r ∈ ω, and hence

kc−(r)Tirα−ρk n−1 Z (−ir+ )t −1 2 0 −1 (irα−ρ)(H(at0 k)) ≤ kAr k · |e τ(κ(at0 k))Tirα−ρσ(k )e dk| K −1 −irt0 + kAr k · ke Φ(r : at0 )c+(r)Tirα−ρk −1 −1 ≤ kAr k · dω max kτ(κ(at0 k))Tirα−ρσ(k )k + kc+(r)Tirα−ρk . k∈K

Hence the claim on kc−(r)Tirα−ρk now follows from (3.16), Lemma 3.7 and the bound for kc+(r)Tirα−ρk. 3.5. Asymptotic expansion of the matrix coeﬃcients of the complementary series. Fix a parameter (n − 1)/2 < s0 < (n − 1), and recall that 2ρ = (n − 1)α. We apply the results of the previous subsections to the standard representation U (s0−n+1)α = U s0α−2ρ. The following theorem is a key ingredient of the proof of Theorem 3.30.

Theorem 3.17. There exist η0 > 0 and N > 1 such that for any σ, τ ∈ Kˆ , for all t > 2, we have

(s0−n+1)α (s0−n+1)t N N (s0−n+1−η0)t Pτ U (at)Pσ = e c+(rs0 )T(s0−n+1)α+O(dσ ·dτ e ), with the implied constant independent of σ, τ.

Proof. Set rs := −i(s − ρ) ∈ C, for all s ∈ C. In particular, =(rs) < 0 for (n − 1)/2 < s < (n − 1). Fix t > 0, and define Ft : C → EM by sα−2ρ Ft(s) := Pτ U (at)Pσ. By Theorem 3.4, Z −1 (sα−2ρ)H(atk) Ft(s) = τ(κ(atk))Tsα−2ρσ(k )e dk. K As was remarked following Theorem 3.8, for each fixed t > 0, the function Ft(s) is analytic on C. Thus in view of Theorem 3.13, we have, whenever rs ∈ Oσ,τ and =(rs) < 0, (s−n+1)t Ft(s) − e c+(rs)Tsα−2ρ is analytic. (3.18) 22 AMIR MOHAMMADI AND HEE OH ˜ Recall the notation Sσ,τ , that is, Oσ,τ = C − ∪r∈Sσ,τ {±r}, and set Sσ,τ = {s : rs ∈ Sσ,τ }. Define (s−n+1)t Gt(s) = Ft(s) − e c+(rs)Tsα−2ρ.

Indeed the map s 7→ Gt(s) is analytic on {s : =(rs) < 0} − S˜σ,τ . Since ˜ 0 ∪σ0,τ 0∈Kˆ ± Sσ0,τ 0 is countable, we may choose a small circle ω of radius at 0 most 1/2 centered at s0 such that {rs : s ∈ ω } ∩ ∪σ0,τ 0∈Kˆ ± Sσ0,τ 0 = ∅. Observe that the intersection of ω0 and the real axis is contained in the interval ((n − 1)/2, n − 1). Note that there exists η > 0 such that for all s ∈ ω0,

(n − 1) − s0 + η < <(s) < s0 + 1 − η. (3.19) 0 Then Gt(s) is analytic on the disc bounded by ω . Hence by the maximum principle, we get

kGt(s0)k ≤ max kGt(s)k. (3.20) s∈ω0 0 Since ω := {rs : s ∈ ω } ⊂ Oσ,τ , Theorems 3.4 and 3.8 imply that for all s ∈ ω0, we have

(s−n+1)t X −`t n−1 Gt(s) = e ( e Γ`(irs − 2 )c+(rs)Tsα−2ρ) `≥1 −st X −`t n−1 + e ( e Γ`(−irs − 2 )c−(rs)Tsα−2ρ). `≥0

Fixing any t0 > 0, by Lemma 3.9, there exists b0 = b0(t0, ω) > 0 such that for all r ∈ ω,

n−1 N0 N0 `t0 kΓ`(ir − 2 )k ≤ b0 · dσ · dτ · e . (3.21) By Proposition 3.15, for all r ∈ ω,

N2 N2 kc±(r)T±irα−ρk ≤ d1 · dσ · dτ .

P −`(t−t0) Let t > t0 + 1 so that `≥0 e ≤ 2. Then we have for any t > 0 and s ∈ ω0,

X −`t n−1 k e Γ`(irs − 2 )c+(rs)Tsα−2ρk `≥1

N0+N2 N0+N2 −t t0 X −`(t−t0) ≤ ·dσ dτ · bσ,τ · e e e `≥0

t0 N0+N2 N0+N2 −t ≤ 2e · b0 · dσ · dτ e and

X −`t n−1 N0+N2 N0+N2 k e Γ`(−irs − 2 )c−(rs)Tsα−2ρk ≤ 2b0 · dσ · dτ . (3.22) `≥0 EFFECTIVE COUNTING 23

0 We now combine these and the expression for Gt(s), for s ∈ ω and get for all t > t0 + 1,

kGt(s0)k

t0 N0+N2 N0+N2 (<(s)−n)t −<(s)t ≤ 2b0(e + 1)dσ · dτ · max(e + e ) s∈ω0

0 N0+N2 N0+N2 (s0−(n−1)−η)t ≤ b · dσ · dτ e where η > 0 is as in (3.19) and b0 > 0 is a constant independent of σ, τ ∈ Kˆ . (s0−n+1)α (s0−n+1)t Since Pτ U (at)Pσ = e c+(rs0 )T(s0−n+1)α + Gt(s0), this ﬁnishes the proof. By Theorem 3.4, Theorem 3.17 implies:

Theorem 3.23. Let (n − 1)/2 < s0 < (n − 1). There exist η0 > 0 and N > 1 such that for all t ≥ 2 and for any unit vectors vσ ∈ Vσ and vτ ∈ Vτ ,

(s0−n+1)α hU (at)(vσ), vτ i

(s0−n+1)t N N (s0−n+1−η0)t = e hc+(rs0 )T(s0−n+1)α(vσ), vτ i + O(dσ dτ e ), with the implied constant independent of σ, τ, vσ, vτ . 3.6. Decay of matrix coefficients for L2(Γ\G). Let Γ < G be a torsion- free geometrically finite group with δ > (n − 1)/2. By Lax-Phillips [40], Patterson [54] and Sullivan [62], U(1, δ−n+1) occurs as a subrepresentation of L2(Γ\G) with multiplicity one, and L2(Γ\G) does not weakly contain any spherical complementary series U(1, s − n + 1) of parameter s strictly bigger than δ. In particular, δ is the maximum s such that U(1, s − n + 1) is weakly contained in L2(Γ\G). The following proposition then follows from [60, Prop. 3.5] and Theorem 3.23: Proposition 3.24. L2(Γ\G) does not weakly contain any complementary series U(υ, s − n + 1) with υ ∈ Mˆ and s > δ. Definition 3.25 (Spectral Gap). We say L2(Γ\G) has a spectral gap if the following two conditions hold:

(1) there exists n0 ≥ 1 such that the multiplicity of any complementary series U(υ, δ −n+1) of parameter δ occurring in L2(Γ\G) is at most dim(υ)n0 for all υ ∈ Mˆ ; (2) there exists (n − 1)/2 < s0 < δ such that no complementary series 2 with parameter s0 < s < δ is weakly contained in L (Γ\G). We set n0(Γ) and s0(Γ) to be the inﬁma of all n0 and of all s0 satisfying (1) and (2) respectively. The pair (n0(Γ), s0(Γ)) will be referred as the spectral gap data for Γ. In other words, the spectral gap property of L2(Γ\G) is equivalent to the following decomposition: 2 † L (Γ\G) = Hδ ⊕ W (3.26) 24 AMIR MOHAMMADI AND HEE OH

† n0 where Hδ = ⊕υ∈Mˆ m(υ)U(υ, δ − n + 1) with m(υ) ≤ dim(υ) , and no complementary series representation with parameter s0(Γ) < s < δ is weakly contained in W. We recall the strong spectral gap property from Def. 1.1. Theorem 3.27. Suppose that δ > (n − 1)/2 for n = 2, 3 or that δ > n − 2 for n ≥ 4. Then L2(Γ\G) has a strong spectral gap property.

Proof. By the the classification of the unitary dual Gˆ explained in the subsection 3.2, any non-spherical complementary series representation is of the ˆ n−1 form U(υ, s − n + 1) for some υ ∈ M − {1} and s ∈ ( 2 , n − 2) (see [26] and [30]). Together with the aforementioned work of Lax-Phillips on the spherical complementary series representations occurring in L2(Γ\G), this implies the claim. ∞ For Ψ ∈ C (Γ\G), ` ∈ N and 1 ≤ p ≤ ∞, we consider the following Sobolev norm: X Sp,`(Ψ) = kX(Ψ)kp (3.28) where the sum is taken over all monomials X in a fixed basis B of g of order p at most ` and kX(Ψ)kp denotes the L (Γ\G)-norm of X(Ψ). Since we will be using S2,` most often, we will set S` = S2,`. For a unitary G-representation space W and a smooth vector w ∈ W , P S`(w) is defined similarly: S`(w) = kX.wk2 where the sum is taken over all monomials X in B of order at most `.

Proposition 3.29. Fix (n − 1)/2 < s0 < (n − 1). Let W be a unitary representation of G which does not weakly contain any complementary series representation U(υ, s − n + 1) with parameter s > s0 and υ ∈ Mˆ . Then for any > 0, there exists c > 0 such that for any smooth vectors w1, w2 ∈ W and for any t > 0, we have

(s0−n+1+)t |hatw1, w2i| ≤ c ·S`0 (w1) ·S`0 (w2) · e where `0 ≥ 1 depends only on G and K. Proof. This proposition is proved in [35, Proof of Prop. 5.3] for n = 3 (based on an earlier idea of [60]), and its proof easily extends to a general n ≥ 2. In the following two theorems, we assume that Γ is Zariski dense in G and 2 that L (Γ\G) has a spectral gap with the spectral gap data (s0(Γ), n0(Γ)).

Theorem 3.30. There exist η > 0 (depending only on s0(Γ)), and ` ∈ N (depending only on n0(Γ)) such that for any real-valued functions Ψ1, Ψ2 ∈ ∞ Cc (Γ\G) as t → +∞, mBR(Ψ ) · mBR∗ (Ψ ) e(n−1−δ)tha Ψ , Ψ i = 1 2 + O(e−ηtS (Ψ )S (Ψ )). t 1 2 |mBMS| ` 1 ` 2 EFFECTIVE COUNTING 25

Proof. As in (3.26), we write

2 † L (Γ\G) = Hδ ⊕ W

† n0(Γ) where Hδ = ⊕υ∈Mˆ m(υ)U(υ, (δ − n + 1)α) with m(υ) ≤ dim(υ) , and no complementary series representation with parameter s0(Γ) < s is weakly contained in W. For simplicity, set s0 := s0(Γ) and n0 := n0(Γ). we set † ⊥ ∞ 0 ⊥ V = Hδ and V = W. Given Ψ1, Ψ2 ∈ Cc (Γ\G), we write Ψi = Ψi + Ψi , 0 ⊥ † where Ψi and Ψi are the projections of Ψi to Hδ and W respectively. Then by Proposition 3.29, there exist `0 ≥ 1 such that for any > 0,

⊥ ⊥ (s0−n+1+)t hatΨ1 , Ψ2 i = O(S`0 (Ψ1)S`0 (Ψ2)e ). (3.31) † If δ = n − 1 and hence if Hδ = C, it is easy to see that (3.31) ﬁnishes the proof. Now suppose δ < n − 1. For each υ ∈ Mˆ , the K-representation Uυ(δ − n + 1)|K is isomorphic to K the induced representation IndM (υ) and hence by the Frobenius reciprocity, the multiplicity of σ in Uυ(s − n + 1)|K is equal to the multiplicity of υ in σ|M , which is denoted by [σ : υ]. Therefore as a K-module,

U(υ, δ − n + 1)|K = ⊕σ∈Kˆ mυ(σ)σ where mυ(σ) ≤ [σ : υ]. As a K-module, we write

† Hδ = ⊕υ∈Mˆ m(υ)U(υ, (δ − n + 1)α) = ⊕υ∈Mˆ m(υ) ⊕σ∈Kˆ mυ(σ)σ

= ⊕σ∈Kˆ m(σ)σ

P n0+1 where m(σ) ≤ υ∈M,υˆ ⊂σ m(υ)[σ : υ]. Note that m(σ) ≤ dσ for n0 = n0(Γ). For each σ ∈ Kˆ , let Θσ be an orthonormal basis in the K-isotypic com- † ponent, say, Vσ, of Hδ, which is formed by taking the union of orthonormal n0+2 bases of each irreducible component of Vσ. Then #Θσ ≤ dσ . By Theorem 3.23 and our discussion in section 3.2, there exist η0 > 0 and N ∈ N such that for any vσ ∈ Θσ and vτ ∈ Θτ , we have for all t 1,

(δ−n+1)t N N (δ−n+1−η0)t hatvσ, vτ i := c(vσ, vτ )e + O(dσ dτ e ) (3.32) where c(vσ, vτ ) = hc+(rδ)T(δ−n+1)αvσ, vτ i. As Ψ0 = P P hΨ , v iv , we have for each t ∈ , i σ∈Kˆ vσ∈Θσ i σ σ R

0 0 X X hatΨ1, Ψ2i = hΨ1, vσi · hΨ2, vτ i · hatvσ, vτ i σ,τ∈Kˆ vσ∈Θσ,vτ ∈Θτ where the convergence follows from the Cauchy-Schwartz inequality. 26 AMIR MOHAMMADI AND HEE OH

Therefore, by (3.32), 0 0 hatΨ1, Ψ2i   X X (δ−n+1)t =  hΨ1, vσihΨ2, vτ ic(vσ, vτ ) e σ,τ∈Kˆ vσ∈Θσ,vτ ∈Θτ

X X N N (δ−n+1−η0)t + dσ dτ hΨ1, vσihΨ2, vτ iO(e )). σ,τ∈Kˆ vσ∈Θσ,vτ ∈Θτ Set   X X E(Ψ1, Ψ2) :=  hΨ1, vσihΨ2, vτ ic(vσ, vτ ) . σ,τ∈Kˆ vσ∈Θσ,vτ ∈Θτ

By (3.3), there exists ` ≥ `0 (depending only on n0) such that

X N+n0+2 N+n0+2 −` −` dσ dτ c(σ) c(τ) < ∞ (3.33) σ,τ∈Kˆ where c(σ) is as in (3.3). Since for any unit vector v ∈ Vσ, −` |hΨ, vi| c(σ) S`(Ψ), we now deduce that

0 0 (δ−n+1)t (δ−n+1−η0)t hatΨ1, Ψ2i = E(Ψ1, Ψ2)e + O(e S`(Ψ1)S`(Ψ2)). Hence, together with (3.31), it implies that there exists η > 0 such that

(δ−n+1)t (δ−n+1−η)t hatΨ1, Ψ2i = E(Ψ1, Ψ2)e + O(e S`(Ψ1)S`(Ψ2)). On the other hand, by Theorem 2.2,

BR BR∗ (n−1−δ)t m (Ψ1) · m (Ψ2) lim e hatΨ1, Ψ2i = . t→∞ |mBMS|

It follows that the inﬁnite sum E(Ψ1, Ψ2) converges and mBR(Ψ ) · mBR∗ (Ψ ) E(Ψ , Ψ ) = 1 2 . 1 2 |mBMS|

This ﬁnishes the proof.

As ha−tΨ1, Ψ2i = hatΨ2, Ψ1i for Ψi’s real-valued, we deduce the following from Theorem 3.30:

Corollary 3.34. There exist η > 0 and ` ∈ N such that, as t → +∞, mBR∗ (Ψ ) · mBR(Ψ ) e(n−1−δ)tha Ψ , Ψ i = 1 2 + O(e−ηtS (Ψ )S (Ψ )). −t 1 2 |mBMS| ` 1 ` 2 EFFECTIVE COUNTING 27

4. Non-wandering component of Γ\ΓHat as t → ∞ 4.1. Basic setup. Let H be either a symmetric subgroup or a horospherical subgroup of G. For the rest of the paper, we will set K,M,A = {at} in each case as follows. If H is symmetric, that is, H is equal to the group of σ-fixed points for some involution σ of G, up to conjugation and commensurability, H is SO(k, 1)×SO(n−k) for some 1 ≤ k ≤ n−1. Let θ be a Cartan involution of G which commutes with σ and set K to be the maximal compact subgroup fixed by θ. Let G = K exp p be the Cartan decomposition and write g as a direct sum of dσ eigenspaces: g = h ⊕ q where h is the Lie algebra of H and q is the −1 eigenspace for dσ. Let a ⊂ p ∩ q be a maximal abelian subspace and set A = exp a = {at := exp(tY0): t ∈ R} where Y0 is a norm one element in a with respect to the Riemannian metric induced by h, i defined in (3.1). Let M be the centralizer of A in K. If H is a horospherical subgroup of G, we let A = {at} be a one-parameter subgroup of diagonalizable elements so that H is the expanding horospherical subgroup for at. Letting M be the maximal compact subgroup in the centralizer of A, we may assume that the right translation action of at cor- 1 n responds to the geodesic flow on T (H ) = G/M. Let K be the stabilizer 1 n of the base point of the vector in T (H ) corresponding to M. n 1 n In both cases, let o ∈ H and X0 ∈ T (H ) be points stabilized by K and M respectively. Let N + and N − be the expanding and contracting horospherical subgroups of G with respect to at, respectively.

4.2. Measures on gH constructed from conformal densities. Set − + + P := MAN , which is the stabilizer of X0 . Via the visual map g 7→ g , we n have G/P ' ∂(H ). Since G/P ' K/M, we may consider the visual map as a map from G to K/M. In both cases, the restriction of the visual map to H induces a diffeomorphism from H/H ∩ M to its image inside K/M. Letting {µx : x ∈ G/K} be a Γ-invariant conformal density of dimension δµ, we will define an H∩M-invariant measureµ ˜gH on each g ∈ G/H. Setting H¯ = H/(H ∩ M), first consider the measure on gH¯ :

δµβ(gh)+ (o,gh) + dµ˜gH¯ (gh) = e dµo((gh) ).

We denote byµ ˜gH the H ∩ M-invariant extension of this measure on gH, that is, for f ∈ Cc(gH), Z Z Z f(gh) dµ˜gH (gh) = f(ghm)dH∩M (m)dµ˜gH¯ (gh) gH¯ H∩M where dH∩M (m) is the probability Haar measure on H ∩ M. By the Γ-invariant conformality of {µx}, this deﬁnition is independent of n o ∈ H andµ ˜gH is invariant under Γ and hence if Γ\ΓgH is closed,µ ˜gH induces a locally ﬁnite Borel measure µgH on Γ\ΓgH. Recall the Lebesgue density {mx} of dimension n − 1 and the Patterson- Sullivan density {νx} of dimension δ. We normalize them so that |mo| = 28 AMIR MOHAMMADI AND HEE OH

|νo| = 1. We set Haar PS µ˜gH =m ˜ gH andµ ˜gH =ν ˜gH , Haar PS and for a closed orbit Γ\ΓgH, we denote by µgH and µgH the measures on Γ\ΓgH induced by them respectively. Haar Haar Haar Lemma 4.1. For each g ∈ G, dµ˜gH (gh) = dµ˜H (h) and dh := dµ˜H (h) is a Haar measure on H.

Proof. As mo is G-invariant, we have −1 + + (n−1)β + (o,g (o)) + dmo((gh) ) = dmg−1(o)(h ) = e h dmo(h ). Since −1 −1 −1 βh+ (o, g (o)) + β(gh)+ (o, gh) = βh+ (o, g (o)) + βh+ (g (o), h) = βh+ (o, h), we have

Haar (n−1)β(gh)+ (o,gh) + (n−1)β + (o,h) + Haar dµ˜gH (gh) = e dµo((gh) ) = e h dµo(h ) = dµ˜H (h) proving the first claim. The first claim shows that dµ˜H¯ is left H-invariant. Haar Since dH∩M is an H∩M-invariant measure, the product measure dµ˜H (hm) = dµ˜H¯ (h)dH∩M (m) is a Haar measure of H. 4.3. Let Γ be a torsion-free, non-elementary, geometrically finite subgroup of G. For any given compact subset Ω of Γ\G, the goal of the rest of this section is to describe the set

{h ∈ Γ\ΓH : hat ∈ Ω for some t > 0}. For H horospherical, this turns out to be a compact subset. For H symmetric, we will obtain a thick-thin decomposition, and give estimates of the size PS Haar of thin parts with respect to the measures µH and µH (Theorem 4.16). An element γ ∈ Γ is called parabolic if there exists a unique fixed point n of γ in ∂(H ), and an element ξ ∈ Λ(Γ) is called a parabolic fixed point if it is fixed by a parabolic element of Γ. Let Λp(Γ) ⊂ Λ(Γ) denote the set of all parabolic fixed points of Γ. Since Γ is geometrically finite, each parabolic fixed point ξ is bounded, i.e., StabΓ(ξ) acts co-compactly on Λ(Γ) − {ξ}. + + − − Recall the notation g = g(X0) and g = g(X0) . n n n−1 Consider the upper half space model for H : H = {(x, y): x ∈ R , y ∈ n n−1 n R>0}. We set R+ := {(x, y): x ∈ R , y ∈ R>0}, so that ∂(R+) = n−1 {(x, 0) : x ∈ R }. Suppose that ∞ is a parabolic fixed point for Γ. 0 Set Γ∞ := StabΓ(∞) and let Γ (∞) be a normal abelian subgroup of Γ∞ which is of finite index; this exists by a theorem of Biberbach. Let L be a n−1 0 minimal Γ∞-invariant subspace in R . By Prop. 2.2.6 in [10], Γ (∞) acts as translations and cocompactly on L. We note that L may not be unique, but any two such are Euclidean-parallel. The notation dEuc and k · k denote the Euclidean distance and the Eu- n clidean norm in R respectively. Following Bowditch [10], we write for each r > 0: n n C(L, r) := {x ∈ R+ ∪ ∂(R+): dEuc(x, L) ≥ r}. (4.2) EFFECTIVE COUNTING 29

Each C(L, r) is Γ∞-invariant and called a standard parabolic region (or an extended horoball) associated to ξ = ∞.

Theorem 4.3. [10, Prop. 4.4] For any 0 > 0, there exists r0 > 0 such that for any r ≥ r0,

(1) γC(L, r) = C(L, r) if γ ∈ Γ∞; (2) if γ ∈ Γ − Γ∞, dEuc(C(L, r), γC(L, r)) > 0. Corollary 4.4. Suppose that ∞ is a bounded parabolic ﬁxed point for Γ. Then for any suﬃciently large r, the natural projection map n n Γ∞\(C(L, r) ∩ H ) → Γ\H is injective and proper.

Proof. We fix 0 > 0, and let r0 > 0 be as in the above theorem 4.3. Let r > n r0, and set C∞ = C(L, r) ∩ H for simplicity. The injectivity is immediate n from Theorem 4.3(2). Since C∞ is closed in H , so is γC∞ for all γ ∈ Γ. Hence to prove the properness of the map, it is sufficient to show that n if F is a compact subset of H , then F intersects at most finitely many distinct γC∞’s. Now suppose there exists an infinite sequence {γi ∈ Γ} such that γiΓ∞’s are all distinct from each other and F ∩ γiC∞ 6= ∅. Choosing yi ∈ F ∩γiC∞, by Theorem 4.3(2), we have d(yi, yj) ≥ 0 for all i 6= j, which contradicts the assumption that F is compact. 4.4. H horospherical. Theorem 4.5. Let H = N be a horospherical subgroup. Suppose that Γ\ΓNM is closed in Γ\G. For any compact subset Ω of Γ\G, the set Γ\ΓNM ∩ ΩA is relatively compact. Proof. We may assume without loss of generality that the horosphere NK/K n in G/K ' H is based at ∞. Note that Γ∞ ⊂ NM and that the closedness of Γ\ΓNM implies that Γ∞\NM → Γ\G is a proper map. Therefore, if the claim does not hold, there exists a sequence ni ∈ NM which is unbounded modulo Γ∞ such that γiniati → x for some ti ∈ R, γi ∈ Γ and x ∈ G. −1 It follows that, passing to a subsequence, niati (o) → ∞ and d(niati , γi x) → −1 0 as i → ∞. Therefore γi x(o) → ∞ and hence ∞ ∈ Λ(Γ). n Since the image of the horosphere N(o) in Γ\H = Γ\G/K is closed, it follows that ∞ is a bounded parabolic fixed point for Γ by [13]. Therefore Γ∞ acts co-compactly on an r neighborhood of a minimal Γ∞-invariant n n−1 subspace L in ∂(H ) − {∞} = R for some r > 0. Write niati (o) = n−1 (xi, yi) ∈ R × R>0. Since {ni} is unbounded modulo Γ∞, after passing to a subsequence if necessary, we have dEuc(xi,L) → ∞. It follows that for any r,(xi, yi) ∈ C(L, r) for all large i. Since ni is unbounded modulo Γ∞, we get niati = (xi, yi) is unbounded in Γ∞\C(L, r). Thus by Corollary 4.4, niati must be unbounded modulo Γ, which is a contradiction. 30 AMIR MOHAMMADI AND HEE OH

4.5. H symmetric. We now consider the case when H is symmetric. Then n H(o) = H/H ∩ K is a totally geodesic submanifold in G(o) = G/K = H . n We denote by π the canonical projection from G to G/K = H . We set S˜ = H(o). Fixing a compact subset Ω of Γ\G, deﬁne

HΩ := {h ∈ H :Γ\Γhat ∈ Ω for some t > 0} and set S˜Ω = HΩ(o). Lemma 4.6. Let ξ ∈ ∂(S˜). If ξ∈ / Λ(Γ), then there exists a neighborhood U n ˜ of ξ in H such that U ∩ SΩ = ∅.

Proof. Let Ω0 be a compact subset of G such that Ω = Γ\ΓΩ0. If the claim does not hold, then there exist hn ∈ H, γn ∈ Γ and tn > 0 such that

γnhnatn ∈ Ω0 and hn(o) → ξ. Note that {hnat(o): t > 0} denotes the (half) geodesic emanating from π(hn) and orthogonal to S.˜ Since hn(o) converges n to ξ ∈ ∂H , it follows that hnatn (o) → ξ. Now since Ω0 is compact, by passing to a subsequence in necessary, n we may assume γnhnatn → x. As G acts by isometries on H , we get −1 γn (x(o)) → ξ. This implies ξ ∈ Λ(Γ) which is a contradiction. Fix a Dirichlet domain D for H ∩ Γ in S˜ and set

DΩ = D ∩ S˜Ω. (4.7) Corollary 4.8. Assume that the orbit Γ\ΓH is closed in Γ\G. There exist a compact subset Y0 of D and a ﬁnite subset {ξ1, . . . ξm} ⊂ Λp(Γ) ∩ ∂(S˜) such that m DΩ ⊂ Y0 ∪ (∪i=1U(ξi)) n ˜ where U(ξi) is a neighborhood of ξi in H . In particular if Λp(Γ)∩∂(S) = ∅, then DΩ is relatively compact. Proof. For each ξ ∈ ∂(S˜) ∩ ∂(D), let U(ξ) be a neighborhood of ξ in Hn. When ξ∈ / Λ(Γ), we may assume by Lemma 4.6 that U(ξ) ∩ S˜Ω = ∅. By the compactness of ∂(S˜) ∩ ∂(D), there exists a ﬁnite covering ∪i∈I U(ξi). Set Y0 := D − ∪i∈I U(ξi), which is a compact subset. Now DΩ − Y0 ⊂

∪i∈I,ξi∈Λ(Γ)U(ξi) by the choice of U(ξi)’s. On the other hand, by [52, Propo- sition 5.1], we have Λ(Γ) ∩ ∂D ⊂ Λp(Γ). Hence the claim follows.

In the rest of this subsection, we ﬁx ξ ∈ Λp(Γ) ∩ ∂(S˜), and investigate n DΩ ∩ U(ξ). We consider the upper half space model for H and assume 0 that ξ = ∞. In particular, S˜ is a vertical plane. Let Γ∞,Γ (∞), L and C(L, r) be as in the subsection 4.3. Without loss of generality, we assume n−1 ⊥ 0 ∈ L. We consider the orthogonal decomposition R = L ⊕ L and let n−1 ⊥ PL⊥ : R → L denote the orthogonal projection map.

Lemma 4.9. There exists R0 > 0 such that for any h ∈ HΩ, we have + kPL⊥ (h )k ≤ R0. EFFECTIVE COUNTING 31

Proof. Let Ω0 be a compact subset of G such that Ω = Γ\ΓΩ0. Then by 0 Corollary 4.4, and part (1) of Theorem 4.3, there exists R0 > 0 depending on Ω, such that 0 n if x ∈ C(L, R0) ∩ H , then x 6∈ ΓΩ0. (4.10)

Suppose now that h ∈ HΩ, thus hat0 (o) ∈ ΓΩ0 for some t0 > 0. This, in 0 0 view of (4.10) and the definition of C(L, R0), implies dEuc(hat0 (o),L) < R0. As discussed above, {hat : t > 0} is the geodesic ray emanating from h(o) and orthogonal to S˜ i.e. a Euclidean semicircle orthogonal to the vertical plane. Hence there exists some absolute constant s0 such that 0 lim dEuc(hat(o),L) ≤ dEuc(hat0 (o),L) + s0 ≤ R0 + s0, t→∞ + 0 which implies kPL⊥ (h )k ≤ R0 := R0 + s0, as we wanted to show. For N ≥ 1, set n n UN (∞) := {x ∈ R+ ∪ ∂(R+): kxkEuc > N}. (4.11) Let ∆ := Γ0(∞) ∩ H and let p be the difference of the rank of Γ0(∞) and the rank of ∆. Suppose p ≥ 1. Let γ = (γ1, . . . , γp) be a p-tuple of elements of Γ0 such that the subgroup generated by γ ∪ ∆ has finite index in Γ0. For p k k1 kp k = (k1, ··· , kp) ∈ Z , we write γ = γ1 ··· γp . The notation |k| means the maximum norm of (k1, ··· , kp). The following gives a description of cuspidal neighborhoods of DΩ: n−1 Theorem 4.12. There exist c0 ≥ 1 and a compact subset F of R such that for all large N 1, + n−1 k {h ∈ R : h ∈ H, π(h) ∈ DΩ ∩ Uc0N (∞)} ⊂ ∪|k|≥N ∆γ F.

Proof. In [52, Prop. 5.8], it is shown that for some c0 ≥ 1 and a compact n−1 subset F of R , + k {h ∈ Λ(Γ) : h ∈ H, π(h) ∈ D ∩ Uc0N (∞)} ⊂ ∪|k|≥N ∆γ F (4.13) for all large N 1. However the only property of h+ ∈ Λ(Γ) used in this + proof is the fact that suph∈H,h+∈Λ(Γ) kPL⊥ (h )k < ∞. Since this property holds for the set in concern by Lemma 4.9, the proof of Proposition 5.8 of [52] can be used. n 4.6. Estimates on the size of thin part. For ξ ∈ ∂(H ), let UN (ξ) be deﬁned to be g(UN (∞)) where g ∈ G is such that ξ = g(∞) and UN (∞) is deﬁned as in (4.11).

Proposition 4.14. Let ξ ∈ ∂(S˜)∩Λp(Γ) and pξ := rank(Γξ)−rank(Γξ ∩H). For all N 1, we have

PS −δ+pξ µ˜H {h ∈ H : π(h) ∈ DΩ ∩ UN (ξ)} N ;

Haar −n+1+pξ µ˜H {h ∈ H : π(h) ∈ DΩ ∩ UN (ξ)} N with the implied constants independent of N. 32 AMIR MOHAMMADI AND HEE OH

Proof. The ﬁrst claim is shown in [52, Proposition 5.2]. Without loss of generality, we may assume that ξ = ∞. By replacing δ by n − 1 and νo by mo in the proof of [52, Proposition 5.2], we get Z (n−1)β + (o,h) + −n+1 e h dmo(h ) |k| h+∈γkF where the notation γ, k and F are as in Theorem 4.12 and f(k) g(k) means that the ratio of f(k) and g(k) lies in between two bounded constants independent of k. Hence by Proposition 4.12,

Haar X −n+1 −n+1+p∞ µ˜H {h ∈ H : π(h) ∈ DΩ ∩ UN (∞)} |k| N . k∈Zp∞ ,|k|≥N Recall the notion of the parabolic-corank of Γ with respect to H, introduced in [52]:

Pb-corankH (Γ) := max (rank(Γξ) − rank(Γξ ∩ H)) . ξ∈Λp(Γ)∩∂(S˜) The following is shown in [52, Thm. 1.14]: Proposition 4.15. We have PS • Pb-corankH (Γ) = 0 if and only if the support of µH is compact; PS • Pb-corankH (Γ) < δ if and only if µH is ﬁnite.

It is also shown in [52, Lem. 6.2] that Pb-corankH (Γ) is bounded above by n − dim(H/(H ∩ K)). Therefore if H is locally isomorphic to SO(k, 1) × PS SO(n − k) and δ > n − k, then µH is ﬁnite. For h ∈ Γ\G, we denote by rh the injectivity radius, that is, the map g 7→ hg is injective on the set d(g, e) ≤ rh. By Corollary 4.8, (4.13), Proposition PS 4.14, and by the structure of the support of µH obtained in [52], we have the following: Theorem 4.16. Suppose that Γ\ΓH is closed. For any compact subset Ω of Γ\G, there exists an open subset YΩ ⊂ Γ\ΓH containing the union PS supp(µH )∪{h ∈ Γ\ΓH : hat ∈ Ω for some t > 0} and satisfying the following properties:

(1) if Pb-corankH (Γ) = 0, YΩ is relatively compact; (2) if Pb-corankH (Γ) ≥ 1, then the following hold: (a) Y := {h ∈ YΩ : rh > } is relatively compact; (b) there exist ξ1, ··· , ξm ∈ Λp(Γ) ∩ ∂(S˜) and c1 > 0 such that for m −1 all small > 0, YΩ − Y ⊂ ∪i=1Uc1 (ξi); (c) for all small > 0,

PS δ−p0 Haar n−1−p0 µH (YΩ − Y) and µH (YΩ − Y)

for p0 := Pb-corankH (Γ). EFFECTIVE COUNTING 33

5. Translates of a compact piece of Γ\ΓH via thickening Let Γ be a non-elementary geometrically ﬁnite subgroup of G. Let H be ± either symmetric or horospherical, and let A = {at}, M, K, N , o, X0 be as in the subsection 4.1. 5.1. Decomposition of measures. Set P := MAN −, which is the stabi- + lizer of X0 . The measure (n−1)β (o,n ) n− 0 − dn0 = e 0 dmo(n0 ) can be seen to be a Haar measure on N − by a similar argument as in Lemma − 4.1. Then for p = n0atm ∈ N AM,

dp := dn0dtdm is a right invariant measure on P where dm is the probability Haar measure of M and dt is the Lebesgue measure on R. For g ∈ G, consider the measure on gP given by δt − dνgP (gp) = e dνo((gp) )dt for t = β(gp)−1 (o, gp). (5.1)

For Ψ ∈ Cc(G), we have: Z Z m˜ Haar(Ψ) = Ψ(gpn)dn dp; (5.2) gP N Z Z BR Haar m˜ (Ψ) = Ψ(gpn)dµ˜gpN (gpn)dνgP (gp); (5.3) gP N Z Z BMS PS m˜ (Ψ) = Ψ(gpn)dµ˜gpN (gpn)dνgP (gp). (5.4) gP N 5.2. Approximations of Ψ. We fix a left invariant metric d on G, which is right H ∩ M-invariant and which descends to the hyperbolic metric on n H = G/K. For a subset S of G and > 0, S denotes the -neighborhood of e in S: S = {g ∈ S : d(g, e) ≤ }. We fix a compact subset Ω of Γ\G. Let r0 := rΩ denote the infimum of the injectivity radius over all x ∈ Ω. That is, for all x ∈ Ω, the map g 7→ xg in injective on the set {g ∈ G : d(g, e) < rΩ}. ∞ We fix a function κΩ ∈ C (Γ\G) such that 0 ≤ κΩ ≤ 1, κΩ(x) = 1 r0 for all x in the 2 -neighborhood of Ω and and κΩ(x) = 0 for x outside the r0-neighborhood of Ω. Fix Ψ ∈ C∞(Ω). For all small > 0, set + − Ψ (x) = sup Ψ(xg) and Ψ (x) = inf Ψ(xg). (5.5) g∈G g∈G

For each 0 < ≤ rΩ, x ∈ Γ\G and g ∈ G, we have − + Ψ (x) ≤ Ψ(xg) ≤ Ψ (x) (5.6) and ± |Ψ (x) − Ψ(x)| ≤ c1S∞,1(Ψ)κΩ(x) 34 AMIR MOHAMMADI AND HEE OH for some absolute constant c1 > 0. For • = Haar, BR, BR∗ or BMS, we deﬁne • • AΨ = S∞,1(Ψ) · m (supp(Ψ)). Deﬁne for each g ∈ G,

φ0(g) = |νg(o)|.

Then φ0 is left Γ-invariant and right K-invariant, and hence induces a ∞ K ∞ n smooth function in C (Γ\G) = C (H ). Moreover φ0 is an eigenfunction of the Laplacian with eigenvalue δ(n − 1 − δ) [63]. Lemma 5.7. For a compact subset Ω of Γ\G, BR Haar (1) m (Ω) supx∈Ω φ0(x) · m (ΩK); BR∗ Haar (2) m (Ω) supx∈Ω φ0(x) · m (ΩK); BMS 2 Haar (3) m (Ω) supx∈Ω φ0(x) · m (ΩK). Proof. The ﬁrst two claims follow since for any K-invariant function ψ in BR∗ BR R Haar Γ\G, m (ψ) = m (ψ) = Γ\G ψ(g)φ0(g)dm (g). The third one follows from the smearing argument of Sullivan, see [63, Proof of Prop. 5]. ∞ On the other hand, there exists ` ∈ N such that for all Ψ ∈ C (Ω), S∞,1(Ψ) S`(Ψ) [1]. Hence it follows from Lemma 5.7 that there exists ∞ ` ∈ N such that for all Ψ ∈ C (Ω), any • = Haar, BR, BR∗ or BMS, and any 0 < < rΩ, • Haar ± AΨ S∞,1(Ψ) · m (supp(Ψ)) S`(Ψ) and S`(Ψ ) S`(Ψ) (5.8) where the implied constants depend only on Ω.

5.3. Thickening of a compact piece of yH. For the rest of this section, fix y ∈ Γ\G and H0 ⊂ H be a compact subset such that the map h 7→ yh is injective on H0. Fix 0 < 0 < rΩ which is smaller than the injectivity radius of yH0. ∞ ∞ 0 Fix non-negative functions Ψ ∈ C (Ω) and φ ∈ C (yH0). Let M ⊂ M be a smooth cross section for H ∩ M in M and set P 0 := M 0AN −. As hp = h0p0 implies h = h0m and p = m−1p0 for m ∈ H ∩ M, it follows that the product map H × P 0 → G is a diffeomorphism onto its image, which is a Zariski open neighborhood of e. Let dp0 be a smooth measure on P 0 such 0 0 ∞ 0 that dp = dH∩M mdp for p = mp . For 0 < < 0, let ρ ∈ C (P) be a R 0 ∞ non-negative function such that ρdp = 1, and we define Φ ∈ Cc (Γ\G) by ( 0 φ(yh)ρ(p) if g = yhp ∈ yH0P Φ(g) = (5.9) 0 otherwise.

Lemma 5.10. For all 0 < < 0 and t > 0, Z Z Z − + Ψ (gat)Φ(g)dg ≤ Ψ(yhat)φ(yh)dh ≤ Ψ (gat)Φ(g)dg. Γ\G h∈H0 Γ\G EFFECTIVE COUNTING 35

0 Proof. For all p ∈ P, h ∈ H0 and t > 0, yhpat = yhat(a−tpat) ∈ yhatP and hence Z Z + Ψ(yhat)φ(yh)dh ≤ Ψ (yhpat)φ(yh)dh. h∈H0 h∈H0

Integrating against ρ, we have Z Ψ(yhat)φ(yh)dh h∈H0 Z + ≤ Ψ (yhpat)φ(yh)ρ(p)dhdp 0 yhp∈yH0P Z + = Ψ (gat)Φ(g)dg. Γ\G

The other direction is proved similarly.

Lemma 5.11. For all 0 < < 0,

BR∗ PS m (Φ) = (1 + O())µyH (φ).

Proof. Choose gy ∈ G so that y = Γ\Γgy and set φ˜(gyh) := φ(yh) and ˜ ˜ 0 Φ(gyhp) := φ(gyh)ρ(p) for hp ∈ H0P and zero otherwise. As 0 < < 0, BR∗ BR∗ ˜ PS PS ˜ we have m (Φ) =m ˜ (Φ) and µyH (φ) =µ ˜yH (φ). By the deﬁnition, we have Z Z BR∗ δβ + (o,g) (n−1)β − (o,g) + − m˜ (Φ˜ ) = Φ˜ (gm)dm e g e g dνo(g )dmo(g )ds g∈G/M M where s = βg− (o, g). For simplicity, we set gy = y ∈ G by abuse of notation. 0 For g = yhp ∈ H0P, as |βg+ (yh, g)| ≤ d(yh, yhp) = d(e, p) ≤ , we have δβ (yh,g) e g+ = 1 + O(). Since g+ = (yh)+, we have

δβg+ (o,g) + δβ(yh)+ (o,yh) + PS e dνo(g ) = (1+O())e dνo((yh) ) = (1+O())dµ˜yH¯ (yh).

On the other hand, as {mx} is G-invariant,

−1 − − (n−1)βp− (o,(yh) (o)) − dmo(g ) = dm(yh)−1(o)(p ) = e dmo(p ).

− − Since p = n0 for p = n0atm, we have −1 βg− (o, g) + βp− (o, (yh) (o)) −1 −1 = βp− ((yh) (o), p) + βp− (o, (yh) (o))

= βp− (o, p)

= β − (o, n0at) = β − (o, at) + β − (o, n0) n0 X0 n0 = −t + β − (o, n0). n0 36 AMIR MOHAMMADI AND HEE OH

−(n−1)t As n0atm ∈ P, we have e = 1 + O() and hence

(n−1)β − (o,g) − e g dmo(g )dsdm −1 (n−1)(β − (o,g)+β − (o,(yh) (o)) − = e g p dmo(p )dsdm (n−1)β (o,n ) −(n−1)t n− 0 − = e e 0 dmo(n0 )dtdm −(n−1)t = e dn0dtdm = (1 + O())dp. 0 0 R ˜ Since dp = dH∩M (m)dp for p = mp , for φ(yh) := H∩M φ(yhm)dH∩M (m), we have Z Z BR∗ ˜ 0 PS 0 m˜ (Φ) = (1 + O()) φ(yh)ρ(p )dµ˜yH¯ (yh)dp 0 P yh∈yH0/(H∩M) PS ˜ = (1 + O())˜µyH (φ). ∞ Corollary 5.12. There exists ` ∈ N such that for any φ ∈ C (yH0), PS µH (φ) S`(φ) where the implied constant depends only on the compact subset yH0.

Proof. By Lemmas 5.11, 5.7 and (5.8), there exists ` ∈ N such that

PS BR∗ µH (φ) m (Φ0 ) S`(Φ`) S`(φ)S`(ρ0 ) S`(φ). where the implied constants depending only on 0 and yH0. Theorem 5.13. Suppose that Γ is Zariski dense in G and that L2(Γ\G) has a spectral gap. Then there exist η0 > 0 and ` ≥ 1 such that for any ∞ ∞ Ψ ∈ C (Ω) and φ ∈ C (yH0), we have Z (n−1−δ)t e Ψ(yhat)φ(yh)dh yh∈yH 1 = mBR(Ψ)˜µPS (φ) + e−η0tO(S (Ψ)S (φ)), |mBMS| yH ` ` with the implied constant depending on Ω and yH0. Proof. It suﬃces to prove the claim for Ψ and φ non-negative. Let ` ≥ 1 be bigger than those `’s in Theorem 3.30, (5.8) and Corollary 5.12. Let q` > 0 0 −q (depending only on the dimension of P ) be such that S`(ρ) = O( ` ), so that −q` S`(Φ) S`(φ)S`(ρ) S`(φ) . ± BR ± BR BR Note that S`(Ψ ) S`(Ψ) and that m (Ψ ) = m (Ψ) + O(AΨ ). By Lemma 5.10, Z − + hatΨ , Φi ≤ Ψ(yhat)φ(yh)dh ≤ hatΨ , Φi. yh∈yH EFFECTIVE COUNTING 37

By Lemma 5.11 and Theorem 3.30, there exists η > 0 such that (n−1−δ)t ± e hatΨ , Φi

1 BR ± BR∗ −ηt −q` = |mBMS| m (Ψ )m (Φ) + e O(S`(Ψ)S`(φ) )

1 BR PS BR PS −ηt −q` = |mBMS| m (Ψ)˜µyH (φ) + O(AΨ µ˜yH (φ)) + e O(S`(Ψ)S`(φ) ). −ηt/(1+q ) By taking = e ` and η0 = η/(1 + q`), we obtain that Z (n−1−δ)t e Ψ(yhat)φ(yh)dh yh∈yH 1 BR PS −η0t BR PS = |mBMS| m (Ψ)˜µyH (φ) + e O(AΨ µ˜yH (φ) + S`(Ψ)S`(φ)). By (5.8) and Corollary 5.12, this proves the theorem. We remark that we don’t need to assume yH is closed in the above theorem, as φ is assumed to be compactly supported. When H is horospherical or symmetric with Pb-corankH (Γ) = 0, Theorem 1.7 is a special case of Theorem 5.13 by Theorem 4.5 and Theorem 4.15.

6. Distribution of Γ\ΓHat and Transversal intersections − Let Γ,H,A = {at}, P = MAN , etc be as in the last section 5. We set + N = N . Let {µx} be a Γ-invariant conformal density of dimension δµ > 0 and letµ ˜gH andµ ˜gN be the measures on gH and gN respectively deﬁned with respect to {µx}.

6.1. Transversal intersections. Fix x ∈ Γ\G. Let 0 > 0 be the injectivity radius at x. In particular, the product map P0 × N0 → Γ\G given by (p, n) 7→ xpn is injective. For any ≤ 0 we set B := PN.

For some c1 > 1, we have N −1 P −1 ⊂ B := PN ⊂ Nc1Pc1 for all c1 c1 > 0. Therefore, in the arguments below, we will frequently identify B with NP, up to a fixed Lipschitz constant. ∞ H∩M ∞ H∩M In the next lemma, let Ψ ∈ Cc (xB0 ) and φ ∈ Cc (yH) . For ± ∞ 0 < ≤ 0, define ψ ∈ C (xP ) by Z ± ± ψ (xp) = Ψ (xpn)dµxpN (xpn) xpN ± where Ψ are as given in (5.5). ± ∞ Define φ ∈ Cc (yH) by + 0 − 0 φ (yh) = sup φ(yhh ) and φ (yh) = inf φ(yhh ). (6.1) 0 0 h ∈H h ∈H Since the metric d on G is assumed to be left G-invariant and right H ∩ −1 −1 M-invariant, we have mHm = H and mNm = N. Therefore the ± ± functions ψ and φ are H ∩ M-invariant. The following lemma is analogous to Corollary 2.14 in [52]; however we 1 n are here working in Γ\G rather than in T (Γ\H ) as opposed to [52]. Let

Px(t) := {p ∈ P0 /(H ∩ M) : supp(φ)at ∩ xpN0 (H ∩ M) 6= ∅}. 38 AMIR MOHAMMADI AND HEE OH

Lemma 6.2. For any 0 < 0, we have Z X − − δµt (1 − c) φ −t (xpa−t)ψ (xp) ≤ e Ψ(yhat)φ(yh)dµyH (yh) ce 0 c yH p∈Px(t) X + + ≤ (1 + c) φ −t (xpa−t)ψ (xp), ce 0 c p∈Px(t) where c > 0 is an absolute constant, depending only on the injectivity radii of supp(φ) and supp(Ψ). Proof. By considering a smooth partition of unity for the support of φ, it suﬃces to prove the lemma, assuming supp(φ) ⊂ yNP ∩ yH ⊂ yB. Fix g, g0 ∈ G so that y = Γg and x = Γg0. Then for H¯ = H/H ∩ M, Z Ψ(yhat)φ(yh)dµyH (yh) yH X Z = Ψ(yhat)φ(yh)dµ˜γgH (γgh) γgH γ∈(Γ∩gHg−1)\Γ X Z Z = Ψ(yhatm) φ(yhm) dmdµ˜γgH¯ (γgh) γgH¯ H∩M γ∈(Γ∩gHg−1)\Γ X Z = Ψ(yhat)φ(yh) dµ˜γgH¯ (γgh) γgH¯ γ∈(Γ∩gHg−1)\Γ as Ψ and φ are H ∩ M-invariant and dm is the probability Haar measure of H ∩ M. Suppose yh ∈ supp(φ) ∩ yH, and write h = nhph where nh ∈ N and + + ph ∈ P. As h = nh and d(h, nh) = O(), we have that for any γ ∈ Γ

dµ˜ ¯ (γgh) γgH = 1 + O(). (6.3) dµ˜γgN (γgnh) −1 0 0 Let γ ∈ (Γ ∩ gHg )\Γ. If γghat = g ph,tnh,t ∈ g P0 N0 , then we claim that

dµ˜ (γgn ) eδµt γgN h = O(e). (6.4) 0 0 dµ˜g ph,tN (g ph,tnh,t) 0 0 −1 Note that γghat = g ph,tnh,t implies γgnhat = g ph,tnh,t(at phat). Hence + 0 + 0 −1 ξ := (γgnh) = (g ph,tnh,t) , and for ph,t := (at phat) ∈ P, 0 0 0 0 βξ(o, γgnh) = βξ(o, g ph,tnh,t) + βξ(g ph,tnh,t, g ph,tnh,tph,t) 0 0 0 0 0 + βξ(g ph,tnh,tph,t, g ph,tnh,tph,ta−t) = βξ(o, g ph,tnh,t) + O() − t, proving the claim (6.4). Note that xB is the disjoint union ∪ xpN . Since n ∈ N and 0 p∈P0 0 h,t 0 0 −t γgh = g ph,ta−t(atnh,ta−t) with atnh,ta−t ∈ Ne 0 , in view of (6.3) and (6.4), EFFECTIVE COUNTING 39 we have Z δµt e Ψ(yhat)φ(yh)dµ˜γgH¯ (γgh) γgH¯ Z X + + 0 = (1 + O()) φce−t (xpa−t) · Ψc(xpn)dµ˜g0pN (g pn) 0 0 p g pN X + + = (1 + O()) φ −t (xpa−t) · ψ (xp) ce 0 c p where the both sums are taken over the set of p ∈ P0 /(H ∩ M) such that 0 γgHat ∩ g pN0 (H ∩ M) 6= ∅ and c > 0 is an absolute constant. Summing over γ ∈ (Γ ∩ gHg−1)\Γ, we obtain one side of the inequality − and the other side follows if one argues similarly using Ψc. By a similar argument, we can prove the following: In the following H∩M ∞ H∩M lemma, let φ ∈ Cc(yH) and ψ ∈ C (xP0 ) , and assume that ∞ µxpN (xpN0 ) > 0 for all p ∈ P0 , so that the function Ψ ∈ C (xB0 ) can be deﬁned by 1 Ψ(xpn) = ψ(xp) µxpN (xpN0 ) for each pn ∈ P0 N0 .

Lemma 6.5. There exists c > 1 such that for all small 0 < ≤ 0, Z − − −δµt X (1 − c) Ψ (yhat)φ −t (yh)dµyH (yh) ≤ e ψ(xp)φ(xpa−t) c ce 0 yH p∈Px(t) Z + + ≤ (1 + c) Ψ (yhat)φ −t (yh)dµyH (yh) c ce 0 yH • Similarly to the deﬁnitions of AΨ, we deﬁne for φ ∈ C(yH) and ψ ∈

C(xP0 ), PS PS ν Aφ := S∞,1(φ) · µyH (supp(φ)),Aψ := S∞,1(ψ) · νxP (supp(ψ)) where νxP is deﬁned as in (5.1). PS ν By a similar argument as in (5.8), we have Aφ S`(φ) and Aψ ≤ S`(ψ) for some ` ∈ N. H∩M ∞ H∩M Lemma 6.6. Let ψ ∈ C(xP0 ) . For Ψ ∈ C (xB0 ) given by 1 Ψ(xpn) = Haar ψ(xp), we have µxpN (xpN0 ) BR BR ν m (Ψ) = νxP (ψ) and AΨ Aψ. − − Proof. For g = xpn, we have g = (xp) and β(xp)− (o, xpn) = β(xp)− (o, xp). Based on this, the claims follow from the deﬁnition. The second claim follows BR from m (supp(Ψ)) = νxP (supp(ψ)) and S∞,1(Ψ) 0 S∞,1(ψ). In the rest of this section, we assume that Γ is Zariski dense and L2(Γ\G) has a spectral gap. 40 AMIR MOHAMMADI AND HEE OH

Theorem 6.7. There exist β > 0 and ` ∈ N such that for any 0 < 0 ∞ H∩M ∞ H∩M and any ψ ∈ C (xP0 ) , and φ ∈ Cc (yH) , we have X 1 e−δt ψ(xp)φ(xpa ) = ν (ψ)µPS (φ) + e−βtO(S (ψ)S (φ)), −t |mBMS| xP yH ` ` p∈Px(t) where Px(t) := {p ∈ P0 /(H ∩ M) : supp(φ)at ∩ xpN0 (H ∩ M) 6= ∅} and the implied constant depends only on the injectivity radii of supp(ψ) and supp(φ). Proof. Deﬁne ± 1 ± Ψ (xpn) = Haar ψ (xp). µxpN (xpN) BR ± ± Then m (Ψ ) = νxP (ψ ) by Lemma 6.6. We take ` big enough to satisfy Theorem 5.13, Corollary 5.12 and that BR ν PS AΨ Aψ S`(ψ) and Aφ S`(φ). By Theorem 5.13, for some η0 > 0, Z (n−1−δ)t ± ± e Ψ (yhat)φ −t (yh)dh e 0 yH 1 BR ± PS ± −η0t ± ± = BMS m (Ψ )µ (φ −t ) + e O(S`(Ψ )S`(φ −t )) |m | yH e 0 e 0 BR PS −t BR PS −η0t = m (Ψ)µyH (φ) + O(( + e )AΨ Aφ ) + e O(S`(Ψ)S`(φ))

PS −η0t = νxP (ψ)µyH (φ) + O((e + )S`(ψ)S`(φ)).

Therefore the claim now follows by applying Lemma 6.5 for dµyH (yh) = −η t/2 dh and δµ = n − 1 with β = η0/2 and = e 0 . Using Theorem 6.7, we now prove the following theorem, which is analo- PS Haar gous to Theorem 5.13 with dh replaced by dµyH (yh). Translates of dµyH PS and dµyH on yH are closely related as their transversals are essentially the same. More precisely, Theorem 6.7 provides a link between translates of these two measures.

Theorem 6.8. There exist β > 0 and ` ∈ N such that for any Ψ ∈ ∞ H∩M ∞ H∩M C (xB0 ) and φ ∈ Cc (yH) , Z PS 1 BMS PS −βt Ψ(yhat)φ(yh)dµyH (yh) = BMS m (Ψ)µyH (φ)+O(e S`(Ψ)S`(φ)). yH |m | ∞ H∩M Proof. Deﬁne ψ ∈ C (xP0 ) by Z PS ψ(xp) = Ψ(xpn)dµxpN (xpn). xpN0 We apply Theorem 6.7 and Lemma 6.2 for the Patterson-Sullivan density {µx} and with this ψ. It follows from the deﬁnition of ψ (see (5.4)) that BMS ν νxP (ψ) = m (Ψ) and Aψ S`(Ψ) for some ` ≥ 1. We take ` large enough to satisfy Theorem 6.7. EFFECTIVE COUNTING 41

Let β be as in Theorem 6.7 and let = e−βt. Now by Lemma 6.2, we get Z PS −δt X ± ± Ψ(yhat)φ(yh)dµ (yh) = (1 + O())e φ −t (xpa−t)ψ (xp). yH e 0 yH p∈Px(t) By Theorem 6.7, −δt X ± ± e φ −t (xpa−t)ψ (xp) e 0 p∈Px(t) 1 ± PS ± −βt = BMS νxP (ψ )µ (φ −t ) + e O(S`(ψ)S`(φ)) |m | yH e 0 1 PS −βt = |mBMS| νxP (ψ)µyH (φ) + O( + e )(S`(ψ)S`(φ)). BMS ν Since νxP (ψ) = m (Ψ) and S`(ψ),Aψ S`(Ψ), this ﬁnishes the proof.

6.2. Effective equidistribution of Γ\ΓHat. We now extend Theorems 5.13 and 6.8 to bounded functions φ ∈ C∞((Γ ∩ H)\H) which are not nec- essarily compactly supported. Hence the goal is to establish the following: set ΓH := Γ ∩ H. PS Theorem 6.9. Suppose that Γ\ΓH is closed and that |µH | < ∞. There exist β > 0 and ` ≥ 1 such that for any compact subset Ω ⊂ Γ\G, for any ∞ ∞ Ψ ∈ C (Ω) and any bounded φ ∈ C (ΓH \H), we have, as t → +∞, Z PS (n−1−δ)t µH (φ) BR −βt e Ψ(hat)φ(h)dh = BMS m (Ψ)+O(e S`(Ψ)S`(φ)) h∈ΓH \H |m | where the implied constant depends only on Ω. We first prove the following which is an analogous version of Theorem 6.9 PS for µH : PS Theorem 6.10. Suppose that Γ\ΓH is closed and that |µH | < ∞. There exist β0 > 0 and ` ≥ 1 such that for any compact subset Ω ⊂ Γ\G, for any Ψ ∈ C∞(Ω)H∩M and for any bounded φ ∈ C∞((Γ ∩ H)\H)H∩M , we have Z µPS(φ) PS H BMS −β0t Ψ(hat)φ(h)dµH (h) = BMS m (Ψ) + O(e S`(φ)S`(Ψ)). h∈ΓH \H |m | PS Proof. Fix ` ∈ N large enough to satisfy Theorem 6.8, Aφ S`(φ) and BMS AΨ S`(Ψ). If H is horospherical, set YΩ = {h ∈ Γ\ΓHM : hat ∈ Ω for some t ∈ R}; if H is symmetric, let YΩ and Y be as in Theorem 4.15 ∞ and set p0 := Pb-corankH (Γ). For > 0, we choose τ ∈ C (YΩ) which is an H ∩ M-invariant smooth approximation of the set Y; 0 ≤ τ ≤ 1, τ(x) = 1 for x ∈ Y and τ(x) = 0 for x∈ / Y/2; we refer to [4] for the construction of −q such τ. Let q` 1 be such that S`(τ) = O( ` ). By the definition of YΩ, R PS we may write the integral Ψ(hat)φ(h)dµ (h) as the sum h∈ΓH \H H Z Z PS PS Ψ(hat)(φ · τ)(h)dµH (h) + Ψ(hat)(φ − φ · τ)(h)dµH (h). ΓH \H YΩ 42 AMIR MOHAMMADI AND HEE OH

PS δ−p0 Note that by (3) of Theorem 4.16, we have µH (YΩ − Y) and PS PS δ−p0 δ−p0 hence µH (φ − φ · τ) Aφ · S`(φ) . Now by Theorem 6.8, Z PS Ψ(hat)(φ · τ)(h)dµH (h) Γ\ΓH PS µ (φ · τ) = H mBMS(Ψ) + O(−q` e−βtS (φ)S (Ψ)) |mBMS| ` ` µPS(φ) = H mBMS(Ψ) + O(APSABMSδ−p0 ) + O(−q` e−βtS (φ)S (Ψ)) |mBMS| φ Ψ ` ` µPS(φ) = H mBMS(Ψ) + O((δ−p0 + −q` e−βt)S (φ)S (Ψ)). |mBMS| ` `

On the other hand,

Z PS Ψ(hat)(φ − φ · τ)(h)dµH (h) YΩ PS δ−p0 S∞,1(Ψ)S∞,1(φ)µH (YΩ − Y) S`(Ψ)S`(φ) .

Hence by combining these two estimates, and taking = e−β/(δ−p0+q`) and −β(δ−p )/(δ−p +q ) β0 := e 0 0 ` , we obtain

Z µPS(φ) PS H BMS −β0t Ψ(hat)φ(h)dµH (h) = BMS m (Ψ) + O(e S`(φ)S`(Ψ)). h∈ΓH \H |m |

Proof of Theorem 6.9. We will divide the integration region into three different regions: compact part, thin part, intermediate range. The compact part is the region where we get the main term using Theorem 5.13. The thin region can be controlled using Theorem 4.16. However there is an intermediate range where we need some control. This is in some sense the main technical diﬀerence from the case where Γ is a lattice. We control the con- tribution from this range, using results proved in this section in particular by relating this integral to summation over the “transversal”; see Lemmas 6.2 and 6.5. We use the notation from the proof of Theorem 6.10. In particular, if H is horospherical, set YΩ = {h ∈ Γ\ΓHM : hat ∈ Ω for some t ∈ R}; if H is symmetric, let YΩ and Y be as in Theorem 4.15 and set p0 :=

Pb-corankH (Γ). Let 0 < 1 < 0. Here, we regard Y0 as a thick part,

Y1 − Y0 as an intermediate range and ΓH \H − Y1 as a thin part. ∞ As above we choose τ0 ∈ C (Y ) which is an H ∩ M-invariant smooth PS δ−p0 approximation of Y0 and recall that µH (YΩ −Y0 ) 0 by (1) of Propo- sition 4.14. EFFECTIVE COUNTING 43

We may write Z Ψ(hat)φ(h)dh h∈ΓH \H Z Z

= Ψ(hat)(φ · τ0 )(h)dh + Ψ(hat)(φ − φ · τ0 )(h)dh. ΓH \H YΩ

Then by Theorem 5.13 with η0 > 0 therein and Theorem 4.16, we get the asymptotic for the thick part:

Z (n−1−δ)t e Ψ(hat)(φ · τ0 )(h)dh ΓH \H µPS(φ) = H mBR(Ψ) + (δ−p0 + e−η0t−q` )O(S (φ)S (Ψ)). (6.11) |mBMS| 0 0 ` `

On the other hand, by Theorem 4.16, we have, for T1 := τ1 − τ0 , Z

Ψ(hat)(φ − φ · τ0 )(h)dh YΩ Z Z !

S∞,1(φ) Ψ(hat)T1 (h)dh + Ψ(hat)dh . (6.12) YΩ YΩ−Y1 R Set Ψ(x) := H∩M Ψ(xm)dm. Applying Lemma 6.2 for the Haar measure Haar PS dµH = dh and Lemma 6.5 for the PS measure µH , and for the function

T := T1 , with the notation as in the proof of Theorem 6.7, we get the following estimate of the integral over the intermediate range Y1 − Y0 : Z (n−1−δ)t e Ψ(hat)T1 (h)dh YΩ Z (n−1−δ)t e Ψ(hat)T1 (h)dh YΩ −δt X + + e ψ (xp)T −t (xpa−t) 1 e 1 p∈Pe(t) Z + + PS Ψ (hat)T −t (h)dµ (h) 1 e 1 H Γ\ΓH

PS δ−p0 S∞,1(Ψ)µH (Y1 − Y0 ) S`(Ψ)0 . (6.13)

Using Theorem 4.16, we also get the following estimate of the integral over the thin part, which is the complement of Y1 : Z (n−1−δ)t (n−1−δ)t n−1+p0 e Ψ(hat)dh ≤ S`(Ψ)e 1 . (6.14) YΩ−Y1 44 AMIR MOHAMMADI AND HEE OH

Therefore by (6.11), (6.12),(6.13), and (6.14), Z (n−1−δ)t e Ψ(hat)φ(h)dh h∈ΓH \H µPS(φ) = H mBR(Ψ) + O(δ−p0 + e−η0t−q` + e(n−1−δ)tn−1−p0 )S (φ)S (Ψ). |mBMS| 0 0 1 ` `

−η0t/(δ−p0+q`) n−1−p0 Recalling δ > p0, take 0 and 1 by 0 = e and 1 = δ−p0 (δ−n+1)t 0 e . We may assume that 1 < 0 by taking ` and hence q` big enough. Finally, we obtain the claim with β := η0(δ − p0)/(δ − p0 + q`).

We can also prove an analogue of Theorem 6.9 with at replaced by a−t, by following a similar argument step by step but using Corollary 3.34 in place PS of Theorem 3.30. Consider the H ∩ M-invariant measure µH,− on ΓH \H δβ (o,h) − induced by the measure e h− dνo(h ) on H¯ = H/(H ∩ M):

PS δβ − (o,h) − dµH,−(hm) = e h dνo(h )dH∩M (m). (6.15) PS Theorem 6.16. Suppose that |µH,−| < ∞. There exist β > 0 and ` ≥ 1 such that for any compact subset Ω in Γ\G, any Ψ ∈ C∞(Ω) and any bounded ∞ φ ∈ C (ΓH \H), we have, as t → +∞, Z µPS (φ) (n−1−δ)t H,− BR∗ −βt e Ψ(ha−t)φ(h)dh = BMS m (Ψ)+O(e S`(Ψ)S`(φ)) h∈ΓH \H |m | where the implied constant depends only on Ω. 6.3. Eﬀective mixing of the BMS measure. In this subsection we prove an eﬀective mixing for the BMS measure:

Theorem 6.17. There exist β > 0 and ` ∈ N such that for any compact subset Ω ⊂ Γ\G, and for any Ψ, Φ ∈ C∞(Ω), Z BMS 1 BMS BMS −βt Ψ(gat)Φ(g)dm (g) = |mBMS| m (Ψ)m (Φ)+O(e S`(Ψ)S`(Φ)) Γ\G with the implied constant depending only on Ω. Proof. Using a smooth partition of unity for Ω, it suﬃces to prove the claim for Φ ∈ Cc(xB0 ) for x ∈ Ω, B0 = P0 N0 and 0 > 0 smaller than the injectivity radius of Ω.

By Theorem 6.8 with H = N and for each p ∈ P0 , Z PS Ψ(xpnat)Φ(xpn)dµxpN (xpn) xpN0 = 1 mBMS(Ψ)µPS (Φ| ) + e−βtO(S (Ψ)S (Φ| )) |mBMS| xpN xpN0 ` ` xpN0 for some β > 0 and ` ∈ N. As Z µPS (Φ| )dν (xp) = mBMS(Φ), xpN xpN0 xP xP0 EFFECTIVE COUNTING 45 we have Z BMS Ψ(gat)Φ(g)dm (g) xB0 Z Z PS = Ψ(xpnat)Φ(xpn)dµxpN (xpn)dνxP (xp) xp∈xP0 xpN0 Z = 1 mBMS(Ψ)mBMS(Φ) + O(e−βt)S (Ψ) · S (Φ| )dν (xp). |mBMS| ` ` xpN0 xP xP0 Since Z S (Φ| )dν (xp) S (Φ)mBR(supp Φ) S (Φ), ` xpN0 xP ` Ω ` xP0 this finishes the proof. 7. Effective uniform counting 7.1. The case when H is symmetric or horospherical. Let G, H, A = {at : t ∈ R}, K, etc be as in the section 5. Let Γ be a Zariski dense and geometrically finite group with δ > (n − 1)/2. Suppose that [e]Γ is discrete PS in H\G, equivalently, Γ\ΓH is closed in Γ\G and that |µH | < ∞. In this section, we will obtain effective counting results from Theorem 6.9 with φ being the constant function 1 on (Γ ∩ H)\H.

Deﬁnition 7.1 (Uniform spectral gap). A family of subgroups {Γi < Γ: i ∈ I} of ﬁnite index is said to have a uniform spectral gap property if

sup s0(Γi) < δ and sup n0(Γi) < ∞. i∈I i∈I where s0(Γi) and n0(Γi) are defined as in (1.3). The pair (supi∈I s0(Γi), supi∈I n0(Γi)) will be referred to as the uniform spectral gap data for the family {Γi : i ∈ I}. As we need to keep track of the main term when varying Γ over its subgroups of finite index for our intended applications to affine sieve, we consider the following situation: let Γ0 < Γ be a subgroup of finite index with Γ0 ∩ H = Γ ∩ H and fix γ0 ∈ Γ. Throughout this section, we assume that both Γ and Γ0 have spectral gaps; hence {Γ, Γ0} is assumed to have a uniform spectral gap. By Theorem 3.27, this assumption is automatic if δ > (n − 1)/2 for n = 2, 3 and if δ > n − 2 for n ≥ 4.

For a family BT ⊂ H\G of compact subsets, we would like to investigate #[e]Γ0γ0 ∩ BT .

Deﬁne a function FT := FBT on Γ0\G by X FT (g) := χBT ([e]γg) γ∈H∩Γ\Γ0 where χBT denotes the characteristic function of BT . Note that

FT (γ0) = #[e]Γ0γ0 ∩ BT . 46 AMIR MOHAMMADI AND HEE OH

Denote by {νx} the Patterson-Sullivan density for Γ normalized so that |νo| = 1. Clearly, {νx} is the unique PS density for Γ0 with |νo| = 1. Recall the Lebesgue density {mx} with |mo| = 1. Therefore ifm ˜ BMS,m ˜ BR andm ˜ Haar are the BMS measure, the BR measure, the Haar measure on G, the corresponding measures mBMS, mBR and Γ0 Γ0 mHaar on Γ \G are naturally induced from them. In particular, for each Γ0 0 • = BMS, BR, Haar, |m• | = [Γ : Γ ] · |m•|. Since H ∩ Γ = H ∩ Γ , we have Γ0 0 0 |µPS| = |µPS | and |µPS | = |µPS |. H Γ0,H H,− Γ0,H,−

∞ 7.2. Weak-convergence of counting function. Fix ψ ∈ Cc (G). For k ∞ k k ∈ K and γ ∈ Γ, deﬁne ψ , ψγ ∈ Cc (G) by ψ (g) = ψ(gk) and ψγ(g) = ∞ ψ(γg). Also deﬁne Ψ, Ψγ ∈ Cc (Γ0\G) by

X 0 X 0 Ψ(g) := ψ(γ g) and Ψγ(g) := ψ(γγ g). 0 0 γ ∈Γ0 γ ∈Γ0

For a function f on K, deﬁne a function ψ ∗K f, or simply ψ ∗ f, on G by Z ψ ∗ f(g) = ψ(gk)f(k) dk. k∈K

± For a subset B ⊂ H\G, deﬁne a function fB on K by Z ± δt fB (k) = e dt. −1 ± a±t∈Bk ∩[e]A

BR BR BR BR∗ We adopt the notationm ˜ + =m ˜ andm ˜ − =m ˜ below. Recall that BMS BMS m means mΓ in the whole section.

Proposition 7.2. There exist β1 > 0 and ` ≥ 1 (depending only on the uniform spectral gap data of Γ and Γ0) such that for any T 1,and any γ ∈ Γ, the pairing hFT , Ψγi in Γ0\G is given by

 PS |µH | BR + (δ−β1)t + BMS m˜ (ψ ∗ f ) + O(maxa ∈B e ·S`(ψ)) if G = HA K  [Γ:Γ0]·|m | BT t T |µPS | P H± BR ± (δ−β1)|t|  BMS m˜ (ψ ∗ f ) + O(maxa ∈B e ·S`(ψ)) otherwise. [Γ:Γ0]·|m | ± BT t T

Proof. For the Haar measure dm˜ Haar(g) = dg, we may write dg = ρ(t)dhdtdk (n−1)|t| −α |t| where g = hatk and ρ(t) = e (1+O(e 1 )) for some α1 > 0 (cf. [52]). |µPS | ± Γ0,H,± ± 1 ± Setting κ (Γ0) := BMS , we have κ (Γ0) = κ (Γ). We will only |m | [Γ:Γ0] Γ0 prove the claim for the case G = HA+K, as the other case can be deduced in a similar fashion, based on Theorem 6.16. We apply Theorem 6.9 and obtain: EFFECTIVE COUNTING 47

Z Z ! hFT , Ψγi = Ψγ(hatk)dh ρ(t)dtdk [e]atk∈BT (H∩Γ)\H Z + (δ−n+1)t BR k = κ (Γ0) e mΓ0 (Ψγ)ρ(t)dtdk [e]atk∈BT Z (δ−n+1−β)t + e ρ(t)dtdk · O(S`(Ψ)) [e]atk∈BT Z + δt BR k (δ−β1)t = κ (Γ0) e m˜ (ψγ )dtdk + O( max e ·S`(ψ)) a ∈B [e]atk∈BT t T where β1 = min{β, α1}. BR BR k BR k By the left Γ-invariance ofm ˜ , we havem ˜ (ψγ ) =m ˜ (ψ ). Hence Z Z Z δt BR k δt BR k BR + e m˜ (ψγ )dtdk = e m˜ (ψ )dtdk =m ˜ (ψ∗fB ). −1 T [e]atk∈BT k∈K at∈BT k This finishes the proof. 1 n 7.3. Counting and the measure MH\G. We denote by X0 ∈ T (H ) the vector fixed by M. In the rest of this section, we define the measures ± dνo (k) on K as follows: for f ∈ C(K), Z Z Z ± ± f(k)dνo (k) = f(km)dmdνo(kX0 ) (7.3) K M\K M where dm is the probability Haar measure of M. Γ Define a measure MH\G = MH\G on H\G: for φ ∈ Cc(H\G),

Observe that the measure MH\G depends on Γ but is independent of the normalization of the PS-density.

Theorem 7.6. If {BT ⊂ H\G} is eﬀectively well-rounded with respect to Γ (see Def. 1.10), then there exists η0 > 0 (depending only on a uniform spectral gap data for Γ and Γ0) such that for any γ0 ∈ Γ

1 1−η0 #([e]Γ0γ0 ∩ B ) = M (B ) + O(M (B ) ) T [Γ:Γ0] H\G T H\G T with the implied constant independent of Γ0 and γ0 ∈ Γ. Proof. Let ψ ∈ C∞(G) be an -smooth approximation of e: 0 ≤ ψ ≤ 1, R + − supp(ψ ) ⊂ G and ψ dg = 1. Set BT, := BT G and BT, := ∩g∈G BT g. Then hFB− , Ψ −1 i ≤ FT (γ0) ≤ hFB+ , Ψ −1 i. T, γ0 T, γ0 48 AMIR MOHAMMADI AND HEE OH

Again, we will provide a proof only for the case G = HA+K; the other case can be done similarly, based on Proposition 7.2. By Proposition 7.2, |µPS | for κ+(Γ ) := Γ0,H , 0 |mBMS| Γ0

+ BR (δ−β1)t −q` hF ± , Ψ i = κ (Γ )m ˜ (ψ ∗ f ± ) + O( max e ). B γ−1 0 B T, 0 T, at∈BT

−q 0 where q` is so that S`(ψ ) = O( ` ). For g = arnk ∈ ANK, deﬁne H(g) = r and κ(g) = k0. Now, using the strong wave front property for ANK decomposition [24], and the deﬁnition 1.10, there exists c > 1 such that for any g ∈ G and T 1, −1 −1 −1 f − (κ(k )) ≤ fBT (κ(k g)) ≤ f + (κ(k )). BT,c BT,c We use the formula form ˜ BR (cf. [52]):

BR −δr − dm˜ (karn) = e dndrdνo (k) and deduce

+ BR κ (Γ)m ˜ (ψ ∗ f + ) BT, Z Z + 0 0 −δr 0 − = κ (Γ) ψ (karnk )fB+ (k )e dk dndrdνo (k) k0∈K KAN T, Z Z + (−δ+(n−1))H(g) − = κ (Γ) ψ (kg)fB+ (κ(g))e dgdνo (k) k∈K G T, Z Z + −1 (−δ+(n−1))H(k−1g) − = κ (Γ) ψ (g)fB+ (κ(k g))e dgdνo (k) k∈K G T, Z Z + −1 − ≤ (1 + O())κ (Γ) ψ (g)fB+ (k )dgdνo (k) k∈K G T,c + p = (1 + O())MH\G(BT,c) = (1 + O( ))MH\G(BT ) (7.7)

R + R −1 − since ψ dg = 1 and κ (Γ) k∈K fBT (k )dνo (k) = MH\G(BT ). Similarly,

+ BR p κ (Γ)m ˜ (ψ ∗ f − ) = (1 + O( ))MH\G(BT ). BT,c

(δ−β1)t 1−η Since maxat∈BT e MH\G(BT ) for some η > 0,

#(Γ γ ∩B ) = 1 M (B )+O(pM (B ))+O(−q` M (B )1−η). 0 0 T [Γ:Γ0] H\G T H\G T H\G T

−η/(p+q`) Hence by taking = MH\G(BT ) and η0 = −pη/(p+q`), we complete the proof.

7.4. Eﬀectively well-rounded families of H\G. EFFECTIVE COUNTING 49

7.4.1. Sectors. For ω ⊂ K, we consider the following sector in H\G:

ST (ω) := [e]{at : 0 ≤ t ≤ log T }ω.

In this subsection, we show that the family of sectors {ST (ω): T 1} is effectively well-rounded provided ω is admissible in the following sense: − −1 Definition 7.8. We will call a Borel subset ω ⊂ K with νo (ω ) > 0 admissible if there exists 0 < p ≤ 1 such that for all small > 0, −1 −1 p νo((ω K − ∩k∈K ω k)) (7.9) with the implied constant depending only on ω. − −1 −1 − Lemma 7.10. Let ω ⊂ K be a Borel subset. If νo (ω ) > 0 and ∂(ω X0 )∩ Λ(Γ) = ∅, then ω is admissible. −1 − Proof. As ∂(ω X0 ) and Λ(Γ) are compact subsets, we can find 0 > 0 −1 − such that the 0-neighborhood of ∂(ω X0 ) is disjoint from Λ(Γ). Hence −1 − we can find 1 > 0 such that ∂(ω )K1 X0 is disjoint from Λ(Γ); so −1 − νo(∂(ω )K1 X0 ) = 0.

Proposition 7.11. Let κo := maxξ∈Λp(Γ) rank (ξ). If n − 2 + κ δ > max{n − 2, 0 }, 2 − −1 −1 then any Borel subset ω ⊂ K such that νo (ω ) > 0 and ∂(ω ) is piece- wise smooth is admissible.

Proof. Let sξ = {ξt : t ∈ [0, ∞)} be a geodesic ray emanating from o toward n ξ and let b(ξt) ∈ H be the Euclidean ball centered at ξ whose boundary is orthogonal to sξ at ξt. Then by Sullivan [63], there exists a Γ-invariant collection of pairwise disjoint horoballs {Hξ : ξ ∈ Λp(Γ)} for which the following holds: there exists a constant c > 1 such that for any ξ ∈ Λ(Γ) and for any t > 0,

−1 −δt d(ξt,Γ(o))(k(ξt)−δ) −δt d(ξt,Γ(o))(k(ξt)−δ) c e e ≤ νo(b(ξt)) ≤ ce e 0 0 where k(ξt) is the rank of ξ if ξt ∈ Hξ0 for some ξ ∈ Λp(Γ) and δ otherwise. Therefore, using 0 ≤ d(ξt, Γ(o)) ≤ t, it follows that for any ξ ∈ Λ(Γ) and t > 1, ( (−2δ+k(ξt))t e if k(ξt) ≥ δ νo(b(ξt)) (7.12) e−δt otherwise.

By standard computations in hyperbolic geometry, there exists c0 > 1 −1 −t −t such that B(ξ, c0 e ) ⊂ b(ξt) ⊂ B(ξ, c0e ) where B(ξ, r) denotes the n Euclidean ball in ∂(H ) of radius r. Hence it follows from (7.12) that if we 0 0 set κ0 := maxξ ∈Λp(Γ) rank (ξ ), then for all small > 0 and ξ ∈ Λ(Γ),

δ 2δ−κ0 νo(B(ξ, )) + . n Clearly, this inequality holds for all ξ ∈ ∂(H ), as the support of νo is equal to Λ(Γ). 50 AMIR MOHAMMADI AND HEE OH

Now if ∂(ω−1) is a piece-wise smooth subset of K, we can cover its - 1−d neighborhood by O( K ) number of -balls, where dK is the dimension of K. Since for any k ∈ K,

+ dM + δ+dM 2δ−κ0+dM νo (B(k, )) · νo(B(k(X0 ), )) + ; where dM is the dimension of M, we obtain that the νo measure of an -neighborhood of ∂(ω−1) is at most of order δ+dM −dK +1 + 2δ−κ0+dM −dK +1 = δ−n+2 + 2δ−κ0−n+2.

n−2+κ0 Hence ω is admissible if δ is bigger than both (n − 2) and 2 .

Corollary 7.13. If δ > n − 2 and rank (ξ) < δ for all ξ ∈ Λp(Γ), then any − −1 −1 Borel subset ω ⊂ K such that νo (ω ) > 0 and ∂(ω ) is piece-wise smooth is admissible. The following strong wave front property of HAK decomposition is a crucial ingredient in proving an effective well-roundedness of a given family: Lemma 7.14 (Strong wave front property). [24, Theorem 4.1] There exists + c > 1 and 0 > 0 such that for any 0 < < 0 and for any g = hatk ∈ HA K with t > 1, gG ⊂ (hHc)(atAc)(kKc) where Hc = H ∩ Gc and Ac and Kc are defined similarly. Proposition 7.15. Let ω ⊂ K be an admissible subset. Then the family {ST (ω): T 1} is effectively well-rounded and PS − −1 |µH |·νo (ω ) δ MH\G(ST (ω)) = δ·|mBMS| (T − 1). Proof. We compute

PS Z log T Z |µH | δt − −1 MH\G(ST (ω)) = |mBMS| e dt dνo (k ) t=0 k∈ω PS − −1 |µH |·νo (ω ) δ = δ·|mBMS| (T − 1). By Lemma 7.14, there exists c ≥ 1 such that for all T > 1 and > 0 + ST (ω)G ⊂ [e]{at : log(1 − c) ≤ t ≤ log(1 + c)T }ωc + where ωc = ωKc and Kc is a c-neighborhood of e in K. Hence with p > 0 given in (7.9),

Hence the family {ST (ω)} is an eﬀectively well-rounded family for Γ.

Therefore we deduce from Theorem 7.6:

Corollary 7.16. Let ω ⊂ K be an admissible subset. Then there exists η0 > 0 (depending only on a uniform spectral gap data for Γ and Γ0) such that for any γ0 ∈ Γ

PS − −1 |µH |·νo (ω ) δ δ−η0 #([e]Γ0γ0 ∩ ST (ω)) = BMS T + O(T ). [Γ:Γ0]·|m |·δ 7.4.2. Counting in norm-balls. Let V be a ﬁnite dimensional vector space on which G acts linearly from the right and let w0 ∈ V . We assume that w0Γ is discrete and that H := Gw0 is either a symmetric subgroup or a horospherical subgroup. We let A = {at},K,M be as in section 5.3. Let λ ∈ N be the log of the largest eigenvalue of a1 on the R-span of w0G, and set ±λ −λt w := lim e w0a±t. 0 t→∞

Fixing a norm k · k on V , let BT := {v ∈ w0G : kvk ≤ T }.

± Proposition 7.17. For any admissible ω ⊂ K, the family {BT ∩ w0A ω} is eﬀectively well-rounded. In particular, {BT } is eﬀectively well-rounded. We also compute that for some 0 < η < δ/λ,

PS Z ± |µH,±| ±λ −δ/λ ± −1 δ/λ η MH\G(BT ∩ w0A ω) = δ·|mBMS| · kw0 kk dνo (k ) · T + O(T ). ω

λ λt λ Proof. By the deﬁnition of λ and w0 , it follows that w0atk = e w0 k + λ t λt O(e 1 ) for some λ1 < λ. Noting that kw0atkk ≤ T implies that e = O(T ) and eλ1t = O(T λ1/λ), we have

+ MH\G(BT ∩ w0A ω)

− for some η < δ/λ. The claim about MH\G(BT ∩ w0A ω) can be proven similarly. To show the eﬀective well-roundedness, we ﬁrst note that by Lemma 7.14, for some c > 1, we have

+ + + (BT ∩ w0A ω)G ⊂ B(1+c)T ∩ w0A ωc. 52 AMIR MOHAMMADI AND HEE OH

Therefore, using the admissibility of ω, and with p given in (7.9), we deduce + + MH\G((BT ∩ w0A ω)G − (BT ∩ w0A ω)) Z Z δt − −1 e dtdνo (k ) + k∈ωc−ω kw0atkk≤(1+c)T Z Z δt − −1 + e dtdνo (k ) + k∈ωc T ≤kw0atkk≤(1+c)T p · T δ/λ + ((1 + c)T )δ/λ − T δ/λ) p δ/λ p + T MH\G(BT ∩ w0A ω). Similarly we can show that + + p + MH\G((BT ∩ w0A ω) − ∩g∈G (BT ∩ w0A ω)g) MH\G(BT ∩ w0A ω). + This ﬁnishes the proof for the eﬀective well-roundedness of {BT ∩ w0A ω}. − The claims about {BT ∩ w0A ω} can be shown in a similar fashion. Put  PS |µH | R λ −δ/λ − −1 +  δ·|mBMS| · K kw0 kk dνo (k ) if G = HA K Ξw0 (Γ) := |µPS | P H± R ±λ −δ/λ ∓ −1  δ·|mBMS| · K kw0 kk dνo (k ) otherwise. We deduce the following from Proposition 7.17 and Theorem 7.6:

Corollary 7.18. (1) For any admissible ω ⊂ K, there exists η0 > 0 such that for any γ0 ∈ Γ,

+ #{v ∈ w0Γ0γ0 ∩ w0A ω : kvk ≤ T }

PS Z |µH | λ −δ/λ − −1 δ/λ δ/λ−η0 = δ·[Γ:Γ ]·|mBMS| · kw0 kk dνo (k )T + O(T ). 0 ω

(2) There exists η0 > 0 (depending only on a uniform spectral gap data for Γ and Γ0) such that for any γ0 ∈ Γ, #{v ∈ w Γ γ : kvk ≤ T } = 1 Ξ (Γ)T δ/λ + O(T δ/λ−η0 ). 0 0 0 [Γ:Γ0] w0 7.5. The case when H is trivial. In this subsection, we will prove the following theorem directly from the asymptotic of the matrix coeﬃcient functions in Theorem 3.30. Γ Recall from the introduction the following Borel measure MG = MG on G: for ψ ∈ Cc(G), Z 1 δt + − −1 MG(ψ) = BMS ψ(k1atk2)e dνo (k1)dtdνo (k2 ). |m | + k1atk2∈KA K

Theorem 7.19. Let Γ0 < Γ be a subgroup of ﬁnite index. If {BT ⊂ G} is eﬀectively well-rounded with respect to Γ (see Def. 1.10), then there exists EFFECTIVE COUNTING 53

η0 > 0 (depending only on a uniform spectral gap data for Γ and Γ0) such that for any γ0 ∈ Γ

#(Γ γ ∩ B ) = 1 M (B ) + O(M (B )1−η0 ) 0 0 T [Γ:Γ0] G T G T with the implied constant independent of Γ0 and γ0 ∈ Γ.

Consider the following function on Γ0\G × Γ0\G: for a compact subset B ⊂ G, X −1 FB(g, h) := χB(g γh)

γ∈Γ0 where χB is the characteristic function of B. We set FT := FBT for simplicity. ± Observe that FT (e, γ0) = #(Γ0γ0 ∩BT ). Let BT, be as in the deﬁnition 1.10 ∞ ∞ and let φ ∈ C (G) and Φ ∈ C (Γ0\G) be as in the proof of Theorem 7.6. We then have

hFB− , Φ ⊗ Φ −1 i ≤ FT (e, γ0) ≤ hFB+ , Φ ⊗ Φ −1 i. T, γ0 T, γ0

Note that for Ψ1, Ψ2 ∈ Cc(Γ0\G) Z Haar 2 hFT , Ψ1 ⊗ Ψ2iΓ0\G×Γ0\G = hΨ1, g.Ψ2iL (Γ0\G) dm (g). g∈BT

For a Borel subset B of G, consider a function fB on K × K given by Z δt fB(k1, k2) = e dt, −1 −1 + at∈k1 Bk2 ∩A and deﬁne a function on G × G by Z −1 ((ψ ⊗ ψ ) ∗ fB)(g, h) = ψ (gk1 )ψ (hk2)fB(k1, k2)dk1dk2. K×K We deduce by applying Theorem 1.4 and using the left Γ-invariance of the measuresm ˜ BR andm ˜ BR that for some η0, η > 0,

hFB, Ψ ⊗ Ψ −1 iΓ0\G×Γ0\G γ0 Z Z Haar = Ψ (g)Ψ −1 (gx)dmΓ0 (g)dx γ0 x∈B Γ0\G Z Z ! −1 Haar (n−1)t −η0t = Ψ (gk1 )Ψ −1 (gatk2)dmΓ0 (g) e (1 + O(e ))dtdk1dk2 γ0 k1atk2∈B Γ0\G 1 Z = eδt(1 + O(e−ηt))mBR(k Ψ )mBR (k−1Ψ)dtdk dk BMS Γ0 2 γ−1 ∗,Γ0 1 1 2 |m | 0 Γ0 k1atk2∈B Z 1 δt −ηt BR BR −1 = e (1 + O(e ))m ˜ (k2ψ )m ˜ (k ψ )dtdk1dk2. |mBMS| ∗ 1 Γ0 k1atk2∈B 54 AMIR MOHAMMADI AND HEE OH

Therefore ± hFB , Φ ⊗ Φ −1 iΓ0\G×Γ0\G (7.20) T, γ0

1 BR∗ BR (δ−η)t q` = (m ˜ ⊗ m˜ )((ψ ⊗ ψ ) ∗ f ± ) + O( max e ). |mBMS| B Γ0 T, at∈BT Recall BR + −δr + − + + dm (karn ) = e dn drdνo (k) for karn ∈ KAN ;

BR∗ − δr − + − − dm (karn ) = e dn drdνo (k) for karn ∈ KAN Haar ± ± and dg = dm˜ (arn k) = drdn dk. For x ∈ G, let κ±(x) denote the K-component of x in AN ±K decomposition and let H±(x) be uniquely given by the requirement x ∈ eH±(x)N ±K. We obtain

BR∗ BR (m ˜ ⊗ m˜ )((ψ ⊗ ψ ) ∗ fB) = Z Z −1 BR∗ BR ψ (g1k1 )ψ (h1k2)fB(k1, k2)dm˜ (g1)dm˜ (h1)dk1dk2 = K×K G×G Z Z − −1 + (δ−n+1)(H−(g)−H+(h)) ψ (kg)ψ (k0h)fB(κ (g) , κ (h))e K×K G×G − + dgdhdνo (k0)dνo (k) =

Z Z − −1 + −1 − −1 −1 + −1 (δ−n+1)(H (gk )−H (hk0 )) ψ (g)ψ (h)fB(κ (gk ) , κ (hk0 ))e K×K G×G − + dgdhdνo (k0)dνo (k); −1 + ﬁrst replacing k1 with k1 , substituting g1 = karn ∈ KAN and h1 = − k0ar0 n0 ∈ KAN and again substituting arnk1 = g and ar0 n0k2 = h. Therefore, using the strong wave front property for AN ±K decomposi- tions [24] and the assumption that R ψdg = 1, we have, for some p > 0,

BR∗ BR (m ˜ ⊗ m˜ )((ψ ⊗ ψ ) ∗ f ± ) BT, Z p −1 + − = (1 + O( )) fBT (k, k0 )dνo (k)dνo (k0) K×K Z p δt + − = (1 + O( )) e dνo (k)dνo (k0) −1 katk0 ∈BT Z p δt + − −1 = (1 + O( )) e dνo (k)dνo (k0 ) katk0∈BT p = (1 + O( ))MG(BT ) with M defined as in Definition 1.8. Since |mBMS| = |mBMS| · [Γ : Γ ], G Γ0 Γ 0 putting the above together, we get 1 1−η0 FT (e, γ0) = MG(BT ) + O(MG(BT ) ) [Γ : Γ0] EFFECTIVE COUNTING 55 for some η0 > 0 depending only on a uniform spectral gap data of Γ and Γ0. This proves Theorem 7.19. −1 Corollary 7.21. Let ω1, ω2 ⊂ K be Borel subsets in K such that ω1 and ω2 are admissible in the sense of (7.8). Set ST (ω1, ω2) := ω1{at : 0 < t < log T }ω2. Then the family {ST (ω1, ω2): T 1} is effectively well-rounded with respect to Γ, and for some η0 > 0, ν+(ω ) · ν−(ω−1) o 1 o 2 δ δ−η0 #(Γ0γ0 ∩ ST (ω1, ω2)) = BMS T + O(T ) δ · |m | · [Γ : Γ0] with the implied constant independent of Γ0 and γ0 ∈ Γ. Using Proposition 7.14 for H = K, we can prove the effective well- roundedness of {ST (ω1, ω2): T 1} with respect to Γ in a similar fashion to the proof of Proposition 7.15. Hence Corollary 7.21 follows from Theorem 7.19; we refer to Lemma 7.10 and Proposition 7.11 for admissible subsets of K.

7.6. Counting in bisectors of HA+K coordinates. We state a counting result for bisectors in HA+K coordinates. ∞ ∞ Let τ1 ∈ Cc (H) with its support being injective to Γ\G and τ2 ∈ C (K), ∞ + and deﬁne ξT ∈ C (G) as follows: for g = hak ∈ HA K, Z −1 ξT (g) = χA+ (a) · τ1(hm)τ2(m k)dm T H∩M + where χ + denotes the characteristic function of A = {at : 0 < t < log T } AT T for T > 1. Since if hak = h0ak0, then h = h0m and k = m−1k0 for some m ∈ H ∩ M, the above function is well-deﬁned.

Theorem 7.22. Let Γ0 < Γ be a subgroup of ﬁnite index. There exist η0 > 0 (depending only on a uniform spectral gap data for Γ and Γ0) and ` ∈ N such that for any γ0 ∈ Γ, µ˜PS(τ ) · ν∗(τ ) X H 1 o 2 δ δ−η0 ξT (γγ0) = BMS T + O(T S`(τ1)S`(τ2)) δ · |m | · [Γ : Γ0] γ∈Γ0 ∗ R − −1 where νo (τ2) := K τ2(k)dνo (k ).

Proof. Deﬁne a function FT on Γ0\G by X FT (g) = ξT (γg).

γ∈Γ0 For any ψ ∈ C∞(G), set Ψ ∈ C∞(Γ \G) to be Ψ(g) = P ψ(γg) and c c 0 γ∈Γ0 then we have: Z Z Z

hFT , ΨiΓ0\G = τ2(k) τ1(h)Ψ(hatk)dh ρ(t)dkdt. + k∈K at∈AT h∈H 56 AMIR MOHAMMADI AND HEE OH

As Ψ ∈ C(Γ0\G), supp(τ1) injects to Γ0\G and H ∩ Γ = H ∩ Γ0, we have µPS (τ ) =µ ˜PS(τ ) and Γ0,H 1 H 1 Z Z τ1(h)Ψ(hatk)dh = τ1(h)Ψ(hatk)dh. h∈H h∈Γ0\Γ0H Therefore, by applying Theorem 5.13 to the inner integral, we obtain η > 0 and ` ∈ N such that

hFT , ΨiΓ0\G PS Z Z µ˜H (τ1) BR δt δ−η = τ2(k)mΓ (Ψk)e dkdt + O(S`(τ1)S`(ψ)T ) |mBMS| + 0 Γ0 k∈K at∈AT PS BR µ˜H (τ1) · m˜ (ψ ∗ τ2) δ δ−η = BMS · T + O(S`(τ1)S`(τ2)S`(ψ)T ). (7.23) δ · |m | · [Γ : Γ0] ,± ,± Let τi be -approximations of τi; τi (x) are respectively the supremum and the infimum of τi in the -neighborhood of x. Then for a suitable ` ≥ 1, PS ,+ ,− − ,+ ,− µ˜H (τ1 − τ1 ) = O( ·S`(τ1)), and νo (τ2 − τ2 ) = O( ·S`(τ2)). ,± Let FT be a function on Γ\G defined similarly as FT , with respect to ,± R ,+ ,+ −1 ξ (hak) = χ + (a) · τ1 (hm)τ2 (m k)dm. T A(1±)T H∩M As before, let ψ ∈ C∞(G) be an smooth approximation of e: 0 ≤ ψ ≤ R 1, supp(ψ ) ⊂ G and ψ dg = 1. Let Ψ −1 be defined as in the subsection γ0 7.2 with respect to ψ. Lemma 7.14 implies that there exists c > 0 such that for all g ∈ Gc, ,− ,+ FT (γ0g) ≤ FT (γ0) ≤ FT (γ0g) and hence ,− ,+ hFT , Ψ −1 i ≤ FT (γ0) ≤ hFT , Ψ −1 i. (7.24) γ0 γ0 By a similar computation as in the proof of Theorem 7.6 (cf. [52, proof of BR ∗ Prop. 7.5]), we havem ˜ (ψ ∗ τ2) = νo (τ2) + O()S`(τ2). −q Therefore, for q` given by S`(ψ ) = O( ` ), we deduce from (7.23) and (7.24) that, using the left Γ-invariance of the measurem ˜ BR,

BMS δ · |m | · [Γ : Γ0] · FT (γ0) PS BR δ δ−η =µ ˜H (τ1) · m˜ (ψ ∗ τ2) · T + O(S`(τ1)S`(τ2)S`(ψ )T )

PS ∗ δ δ −q` δ−η =µ ˜H (τ1)νo (τ2)T + O(T + T )S`(τ1)S`(τ2)

PS ∗ δ δ−η0 =µ ˜H (τ1)νo (τ2)T + O(T )S`(τ1)S`(τ2)

−η/(1+q ) for some η0 > 0, by taking = T ` . Corollary 7.21 as well as its analogues in the HAK decomposition can be deduced easily from Theorem 7.22 by approximation admissible sets by smooth functions. EFFECTIVE COUNTING 57

8. Affine sieve In this ﬁnal section, we prove Theorems 1.16 and 1.17. We begin by recalling the combinatorial sieve (see [25, Theorem 7.4]). Let A = {an} be a sequence of non-negative numbers and let B be a ﬁnite Q set of primes. For z > 1, let P be the product of primes P = p/∈B,p

Ad := {an ∈ A : n ≡ 0(d)} P and set |Ad| := n≡0(d) an. We will use the following combinatorial sieve:

Theorem 8.1. (A1) For d square-free with no factors in B, suppose that

|Ad| = g(d)X + rd(A) where g is a function on square-free integers with 0 ≤ g(p) < 1, g is multiplicative outside B, i.e., g(d1d2) = g(d1)g(d2) if d1 and d2 are square-free integers with (d1, d2) = 1 and (d1d2,B) = 1, and for some c1 > 0, g(p) < 1 − 1/c1 for all prime p∈ / B. (A2) A has level distribution D(X ), in the sense that for some > 0 and C > 0, X 1− |rd(A)| ≤ CX . d

(A3) A has sieve dimension r in the sense that there exists c2 > 0 such that for all 2 ≤ w ≤ z, X z −c ≤ g(p) log p − r log ≤ c . 2 w 2 (p,B)=1,w≤p≤z Then for s > 9r, z = D1/s and X large enough, X S(A,P ) . (log X )r m Let G, GV = C , Γ, w0 ∈ V (Z), etc., be as in Theorem 1.16. We consider the spin cover G˜ → G. Noting that the image of G˜ (R) is precisely ◦ G = G(R) , we replace Γ by its preimage under the spin cover. This does not aﬀect the orbit w0Γ and all our counting statements hold equally. Set W := w0G (resp. w0G ∪ {0}) if w0G (resp. w0G ∪ {0}) is Zariski closed, Let F ∈ Q[W ] be an integer-valued polynomial on w0Γ and let F = F1 ··· Fr where Fi ∈ Q[W ] are all irreducible also in C[W ] and integral on the orbit w0Γ. We may assume without loss of generality that gcd{F (x): −1 x ∈ w0Γ} = 1, by replacing F by m F for m := gcd{F (x): x ∈ w0Γ}. 58 AMIR MOHAMMADI AND HEE OH

Let {BT ⊂ w0G} be an eﬀectively well-rounded family of subsets with respect to Γ. Set O := w0Γ. For n ∈ N, d ∈ N, and T > 1, we also set

an(T ) := #{x ∈ O ∩ BT : F (x) = n};

Γw0 (d) := {γ ∈ Γ: w0γ ≡ w0 (d)}, X |A(T )| := an(T ) = #O ∩ BT ; n X |Ad(T )| := an(T ) = #{x ∈ O ∩ BT : F (x) ≡ 0 (d)}. n≡0(d)

Let Γd := {γ ∈ Γ: γ ≡ e (d)}. Theorem 8.2. If δ > n − 2, then there exists a ﬁnite set S of primes such that the family {Γd : d is square-free with no factors in S} has a uniform spectral gap.

Proof. As δ > (n−1)/2, by [58] and by the transfer property obtained in [6], 2 n there exists a ﬁnite set S of primes such that the family L (Γd\H ) has a uniform spectral gap where d runs over all square-free integers with no prime 2 factors in S, that is, there exists s1 < δ such that L (Γd\G) does not contain a spherical complementary series representation of parameter s1 < s < δ. By 2 Theorem 3.27 and the classiﬁcation of Gˆ [30], L (Γd\G) does not contain a non-spherical complementary series representation of parameter s > (n−2). 2 It follows that L (Γd\G) = Hδ ⊕ Wd where Hδ = U(1, (δ − n + 1)α) is the spherical complementary series representation of parameter δ; hence n0(Γd) = 1 and Wd does not weakly contain any complementary series representation of parameter max(n − 2, s1) < s < δ. So sup s0(Γd) ≤ max(n − 2, s1) < δ and sup n0(Γd) = 1 where d runs over all square-free integers with no prime factors in S.

Denote by Γ(d) the image of Γ under the reduction map G˜ → G˜ (Z/dZ) m and set Od to be the orbit of w0 in (Z/dZ) under Γ(d); so #Od = [Γ :

Γw0 (d)]. We also set

OF (d) := {x ∈ Od : F (x) ≡ 0 (d)}.

Corollary 8.3. Put Mw0G(BT ) = X . Suppose that for some ﬁnite set S of primes, the family {Γd : d is square-free with no factors in S} has a uniform spectral gap. Then there exists η0 > 0 such that for any square-free integer d with no factors in S, we have

|Ad(T )| = g(d)X + rd(A)

#OF (d) 1−η0 where g(d) = and rd(A) = #OF (d) · O(X ). #Od EFFECTIVE COUNTING 59

Proof. Since Γd ⊂ Γw0 (d), the assumption implies that the family {Γw0 (d): d is square-free with no factors in S} has a uniform spectral gap. Therefore,

Theorem 1.12 on #(w0Γw0 (d)γ ∩ BT ) implies that for some uniform 0 > 0, X |Ad(T )| = #(w0Γw0 (d)γ ∩ BT )

γ∈Γw0 (d)\Γ,F (w0γ)≡0(d) X = 1 X + O(X 1−0 ) . [Γ:Γw0 (d)] γ∈Γw0 (d)\Γ,F (w0γ)≡0(d)

Since #OF (d) = #{γ ∈ Γw0 (d)\Γ,F (w0γ) ≡ 0 (d)}, the claim follows.

In the following we verify the sieve axioms (A1), (A2) and (A3) in this setup. This step is very similar to [48, sec. 4] as we use the same combinatorial sieve and the only diﬀerence is that we use the variable X = Mw0G(BT ) instead of T . This is needed for us, as we are working with very general α sets BT ; however if Mw0G(BT ) T for some α > 0, we could also use the parameter T . Using a theorem of Matthews, Vaserstein and Weisfeiler [46], and enlarg- Q ˜ ing S if necessary, the diagonal embedding of Γ is dense in p/∈S G(Fp). The multiplicative property of g on square-free integers with no factors in S follows from this (see [48, proof of Prop. 4.1]). Letting Wj = {x ∈ W : Fj(x) = 0}, Wj is an absolutely irreducible aﬃne variety over Q of dimension dim(W ) − 1 and hence by Noether’s theorem, Wj is absolutely irreducible over Fp for all p∈ / S, by enlarging S if necessary. We may also assume that W (Fp) = w0G(Fp) (possibly after adding {0}) for all p∈ / S by Lang’s theorem [38]. Using Lang-Weil estimate [39] on #W (Fp) and #Wj(Fp), we obtain that for p∈ / S, dim(W )−1 dim W −3/2 dim W dim W −1/2 #OF (p) = r·p +O(p ) and #Op = p +O(p ). Hence g(p) = r · p−1 + O(p−3/2) for all p∈ / S. This implies A3 (cf. [47, Thm 2.7]), as well as the last claim of A1. Moreover this together with Corollary 8.3 imply that r(A, d) ddim W −1X 1−η0 . η /(2 dim W ) Hence for D ≤ X 0 and 0 = η0/2, X r(A, d) Ddim W X 1−η0 ≤ X 1−0 , d≤D

1/s η /(2s dim W ) providing (A2). Therefore for any z = D ≤ X 0 and s > 9r, and for all large X , we have X S(A,P ) . (8.4) (log X )r 60 AMIR MOHAMMADI AND HEE OH

Proof of Theorem 1.16. Using arguments in the proof of Corollary 8.3, we ﬁrst observe (cf. [48, Lem. 4.3]) that there exists η > 0 such that for any k ∈ N, 1−η #{x ∈ O ∩ BT : Fj(x) = k} X . Fixing 0 < 1 < η, it implies that

1 1−η+1 #{x ∈ O ∩ BT : |Fj(x)| ≤ X } X . (8.5) Now r X 1 #{x ∈ O ∩ BT : all Fj(x) prime} ≤ #{x ∈ O ∩ BT : |Fj(x)| ≤ X } j=1 1 + #{x ∈ O ∩ BT : |Fj(x)| ≥ X for all 1 ≤ j ≤ r and all Fj(x) prime}.

η0/(2s dim W ) Q 1 Now for z ≤ X such that P = p

1 {x ∈ O ∩ BT : |Fj(x)| ≥ X for all 1 ≤ j ≤ r and all Fj(x) prime}

⊂ {x ∈ O ∩ BT :(Fj(x),P ) = 1} and the cardinality of the latter set is S(A,P ) according to our deﬁnition of an’s. Therefore, we obtain the desired upper bound: X X #{x ∈ O ∩ B : all F (x) prime} X 1−η+1 + . T j (log X )r (log X )r Proof of Theorem 1.17. By the assumption, for some β > 0, β β max kxk Mw0G(BT ) = X . (8.6) x∈BT It follows that β deg(F ) β deg(F ) max |F (x)| Mw0G(BT ) = X . (8.7) x∈BT β·deg(F )2s dim W Then for z = X η0/(2s dim W ) and P = Q p, R = , we p

{x ∈ O ∩ BT :(F (x),P ) = 1} ⊂

{x ∈ O ∩ BT : F (x) has at most R prime factors}, since all prime factors of F (x) has to be at least the size of z if (F (x),P ) = 1 β deg(F ) and |F (x)| X if x ∈ BT . Since S(A,P ) = #{x ∈ O ∩ BT : X (F (x),P ) = 1}, we get the desired lower bound (log X )r from (8.4). Remark 8.8. When Γ is an arithmetic subgroup of a simply connected semisimple algebraic Q-group G, and H is a symmetric subgroup, the analogue of Theorem 1.12 has been obtained in [4], assuming that H ∩ Γ is a lattice in H. Strictly speaking, [4, Theorem 1.3] is stated only for a ﬁxed group Γ; however it is clear from its proof that the statement also holds uniformly over its congruence subgroups with the correct main term, as in Theorem 1.12. Based on this, one can use the combinatorial sieve 8.1 to EFFECTIVE COUNTING 61 obtain analogues of Theorems 1.16 and 1.17, as it was done for a group variety in [48]. Theorem 1.17 on lower bound for Γ arithmetic was obtained in [21] further assuming that H ∩ Γ is co-compact in H.

Acknowledgment: We are grateful to Gregg Zuckerman for pointing out an important mistake regarding the spectrum of L2(Γ\G) in an earlier version of this paper. We would like to thank Peter Sarnak for helpful comments and Sigurdur Helgason for the reference on the work of Harish-Chandra. We also thank Dale Winter for helpful conversations on related topics.

References [1] T. Aubin. Nonlinear analysis on manifolds. GM 252 Springer, 1982. [2] Martine Babillot. On the mixing property for hyperbolic systems. Israel J. Math., 129:61–76, 2002. [3] M. W. Baldoni Silva and D. Barbasch The unitary spectrum for real rank one groups. Inventiones Math., 72:27–55, 1983. [4] Yves Benoist and Hee Oh. Effective equidistribution of S-integral points on symmetric varieties. Annales de L’Institut Fourier, Vol 62 (2012), 1889-1942. [5] Jean Bourgain, Alex Gamburd, and Peter Sarnak. Generalization of Selberg’s 3/16 theorem and Affine sieve. Acta Math. 207, (2011) 255-290. [6] Jean Bourgain, Alex Gamburd, and Peter Sarnak. Affine linear sieve, expanders, and sum-product. Inventiones 179, (2010) 559-644. [7] Jean Bourgain, Alex Kontorovich, and Peter Sarnak. Sector estimates for Hyperbolic isometries. GAFA 20, 1175–1200, 2010. [8] Jean Bourgain and Alex Kontorovich. On representations of integers in thin subgroups of SL(2, Z). GAFA 20, 1144–1174, 2010. [9] Jean Bourgain and Alex Kontorovich. On the local-global conjecture for integral Apollonian gaskets. Inventiones, Vol 196, 2014, 589-650 [10] B. H. Bowditch. Geometrical finiteness for hyperbolic groups. J. Funct. Anal., 113(2):245–317, 1993. [11] Rufus Bowen. Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc., 154:377–397, 1971. [12] Marc Burger. Horocycle flow on geometrically finite surfaces. Duke Math. J., 61(3):779–803, 1990. [13] F. Dal’bo. Topologie du feuilletage fortement stable. Ann. Inst. Fourier (Grenoble), 50(3):981–993, 2000. [14] Dmitry Dolgopyat. On decay of correlations in Anosov flows. Ann. Math, 147:357–390, 1998. [15] J. Diximer. Sur les représentations de certains groupes orthogonaux. C. R. Acad. Sc. Paris, vol 89 (1960), 3263-3265. [16] W. Duke, Z. Rudnick, and P. Sarnak. Density of integer points on affine homogeneous varieties. Duke Math. J., 71(1):143–179, 1993. [17] Alex Eskin and C. T. McMullen. Mixing, counting, and equidistribution in Lie groups. Duke Math. J., 71(1):181–209, 1993. [18] L. Flaminio and R. Spatzier. Geometrically finite groups, Patterson-Sullivan measures and Ratner’s theorem. Inventiones, 99 (1990), 601-626. [19] Alex Gamburd. On the spectral gap for infinite index congruence subgroups of SL2(Z). Israel J. Math, 127 (2002), 157-200. [20] Alex Gorodnik and Amos Nevo. The ergodic theory of Lattice subgroups. Annals of Math Studies. Vol 172, 2009. 62 AMIR MOHAMMADI AND HEE OH

[21] Alex Gorodnik and Amos Nevo. Lifting, Restricting and Sifting integral points on affine homogeneous varieties. To appear in Compositio Math. arXiv:1009.5217. [22] Alex Gorodnik and Hee Oh. Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary. Duke Math. J., 139(3):483-525, 2007. [23] Alex Gorodnik, Hee Oh and Nimish Shah. Integral points on symmetric varieties and Satake compactifications American J. Math., 131:1-57, 2009. [24] Alex Gorodnik, Hee Oh and Nimish Shah. Strong wavefront lemma and counting lattice points in sectors. Israel J. Math., 176:419-444, 2010. [25] H. Halberstam and H. Richert. Sieve methods. Academic Press., (1974) 167-242. [26] Takeshi Hirai. On irreducible representations of the Lorentz group of n-th order. Proc. Japan. Acad., Vol 38., 258-262, 1962. [27] Roger Howe and Calvin Moore. Asymptotic properties of unitary representations. J. Funct. Anal., 72-96, 1979. [28] Henryk Iwaniec and Emmanuel Kowalski. Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004. [29] D. Y. Kleinbock and G. A. Margulis. Bounded orbits of nonquasiunipotent flows on homogeneous spaces. In Sina˘ı’sMoscow Seminar on Dynamical Systems, volume 171 of Amer. Math. Soc. Transl. Ser. 2, pages 141-172. Amer. Math. Soc., Providence, RI, 1996. [30] A. Knapp and E. Stein. Intertwining operators for semisimple groups Annals of Math., 489-578, 1971. [31] Inkang Kim. Counting, Mixing and Equidistribution of horospheres in geometrically finite rank one locally symmetric manifolds. To appear in J. Reine Angew. Math., arXiv:1103.5003. [32] A. Knapp. Representation theory of semisimple Lie groups. Princeton University press. [33] Alex Kontorovich. The hyperbolic lattice point count in infinite volume with applications to sieves. Duke Math. J., 149(1):1-36, 2009. [34] Alex Kontorovich and Hee Oh. Almost prime Pythagorean triples in thin orbits. J. Reine Angew. Math. Vol 667, 89-131, 2012. [35] Alex Kontorovich and Hee Oh. Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds. Journal of AMS., Vol 24 (2011), 603-648. [36] E. Kowalski. Sieve in expansion. Séminaire Bourbaki arXiv:1012.2793. [37] E. Kowlaski. Sieve in discrete groups, especially sparse. In ”Thin groups and superstrong approximation”, edited by Breuillard and Oh, Math. Sci.Res.Inst.Publ. 61, Cambridge Univ. Press. Cambridge 2014 [38] S. Lang. Algebraic groups over finite fields. American J. Math. 78 (1956) 555-563. [39] S. Lang and A. Weil. Number of points of varieties in finite fields. American J. Math. 76 (1954) 819-827. [40] Peter D. Lax and Ralph S. Phillips. The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces. J. Funct. Anal., 46(3):280-350, 1982. [41] Min Lee and Hee Oh. Effective count for Apollonian circle packings and closed horospheres. GAFA., Vol 23 (2013), 580-621. [42] J. Liu and P. Sarnak. Integral points on quadrics in three variables whose coordinates have few prime factors. Israel J. Math., 178 (2010) 393-426. [43] M. Magee. Quantitative spectral gap for thin groups of hyperbolic isometries. JEMS, Vol 17 ( 2015), 151-187 [44] F. Maucourant. Homogeneous asymptotic limits of Haar measures of semisimple Lie groups and their lattices. Duke Math. J.. vol 136 (2) 1007, 357-399. [45] Gregory Margulis. On some aspects of theory of Anosov systems. Springer Mono- graphs in Mathematics. Springer-Verlag, Berlin, 2004. EFFECTIVE COUNTING 63

[46] C. Matthews, L. Vaserstein and B. Weisfeiler. Congruence properties of Zariski dense subgroups. Proc. London Math. Soc. 48, 1984, 514-532. [47] H. Montgomery and R. Vaughan. Multiplicative number theory. I. Classical theory. Camb. Studies in Advanced Math. 97, Camb, Univ. Press. 2007. [48] Amos Nevo and Peter Sarnak. Prime and Almost prime integral points on principal homogeneous spaces. Acta Math., 205 (2010), 361-402 [49] Hee Oh. Orbital counting via mixing and unipotent flows Homogeneous flows, Moduli spaces and Arithmetic. Clay Math. Proceedings (2010), Vol 10, 339-375. [50] Hee Oh. Dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond. Proceedings of ICM. 2010, Vol III, 1308-1331. [51] Hee Oh. Harmonic analysis, Ergodic theory and Counting for thin groups. In ”Thin groups and superstrong approximation”, edited by Breuillard and Oh, Math. Sci.Res.Inst.Publ. 61, Cambridge Univ. Press. Cambridge 2014 [52] Hee Oh and Nimish Shah. Equidistribution and counting for orbits of geometrically finite hyperbolic groups. Journal of AMS. Vol 26 (2013), 511-562. [53] Hee Oh and Nimish Shah. The asymptotic distribution of circles in the orbits of Kleinian groups. Inventiones, Vol 187, 1-35, 2012. [54] S.J. Patterson. The limit set of a Fuchsian group. Acta Mathematica, 136:241–273, 1976. [55] J. Parkkonen and F. Paulin. Counting common perpendicular arcs in negative curvature. Preprint, arXiv:1305.1332. [56] Thomas Roblin. Ergodicitéet équidistributionen courbure négative. Mém.Soc. Math. Fr. (N.S.), (95):vi+96, 2003. [57] A. Salehi Golsefidy and P. Sarnak. Affine Sieve. Journal of AMS, Vol 26, (2013), 1085-1105 [58] A. Salehi Golsefidy and P. Varjú.Expansion in perfect groups. GAFA. Vol 22 (2012), 1832-1891. [59] Barbara Schapira. Equidistribution of the horocycles of a geometrically finite surface. Int. Math. Res. Not., (40):2447-2471, 2005. [60] Yehuda Shalom. Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group. Ann. of Math. (2), 152(1):113-182, 2000. [61] L. Stoyanov. Spectra of Ruelle transfer operators for axiom A flows. Nonlinearity, 24 (2011), 1089-1120. [62] Dennis Sullivan. The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Etudes´ Sci. Publ. Math., (50):171-202, 1979. [63] Dennis Sullivan. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math., 153(3-4):259-277, 1984. [64] Ilya Vinogradov. Effective bisector estimate with applications to Apollonian circle packings. IMRN, Vol. 2014, No 12, 3217-3262 [65] Garth Warner. Harmonic Analysis on semisimple Lie groups I. Mém.Soc. Math. Fr. (N.S.), (95):vi+96, 2003. [66] Garth Warner. Harmonic Analysis on semisimple Lie groups II. Mém.Soc. Math. Fr. (N.S.), (95):vi+96, 2003. [67] Dale Winter. Mixing of frame flow for rank one locally symmetric spaces and measure classification. To appear in Israel Journal of Math, arXiv 1403.2425. 64 AMIR MOHAMMADI AND HEE OH

Department of Mathematics, The University of Texas at Austin, Austin, TX 78750 E-mail address: [email protected]

Mathematics department, Yale university, New Haven, CT 06520 and Korea Institute for Advanced Study, Seoul, Korea E-mail address: [email protected]